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Your data matches 84 different statistics following compositions of up to 3 maps.
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Matching statistic: St001114
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
St001114: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 0
[1,3,4,2] => 1
[1,4,2,3] => 0
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 0
[2,3,4,1] => 1
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 2
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 1
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 1
[1,2,4,5,3] => 0
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 1
[1,4,3,5,2] => 0
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
Description
The number of odd descents of a permutation.
Matching statistic: St001153
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001153: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001153: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [1,1,0,0]
=> {{1,2}}
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 0
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 0
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 0
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 0
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 0
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 0
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 0
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 0
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 0
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 0
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1
Description
The number of blocks with even minimum in a set partition.
Matching statistic: St000260
(load all 48 compositions to match this statistic)
(load all 48 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 75%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 75%
Values
[1,2] => [1,2] => [1,2] => ([],2)
=> ? = 0
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0}
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0}
[2,3,1] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,0}
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2}
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2}
[1,3,4,2] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2}
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2}
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,3,1,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2}
[2,3,4,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2}
[3,1,4,2] => [2,1,4,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2}
[3,2,4,1] => [2,1,4,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,2,1] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2}
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [4,2,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2}
[4,2,3,1] => [4,1,3,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2}
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,2,4,5,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,3,4,5,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,4,2,5,3] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,4,3,5,2] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,3,2] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,3,2,4] => [1,5,3,2,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,3,4,2] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,4,2,3] => [1,5,4,2,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,1,3,5,4] => [2,1,3,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,1,4,5,3] => [2,1,3,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[2,1,5,3,4] => [2,1,5,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3,1,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,3,1,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[2,3,4,1,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,3,4,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,3,5,1,4] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3,5,4,1] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,4,1,3,5] => [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,4,1,5,3] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[2,4,3,1,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,4,3,5,1] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[2,4,5,1,3] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,4,5,3,1] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,5,1,3,4] => [2,5,1,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,5,1,4,3] => [2,5,1,4,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,3,1,4] => [1,5,3,2,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,5,3,4,1] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,5,4,1,3] => [2,5,4,1,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,5,4,3,1] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,1,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,1,2,5,4] => [3,1,2,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,1,4,2,5] => [2,1,4,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,2,4,1,5] => [2,1,4,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,4,1,2,5] => [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,4,2,1,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,5,1,2,4] => [3,5,1,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[4,1,2,3,5] => [4,1,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[4,1,2,5,3] => [3,1,2,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[4,1,3,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[4,1,3,5,2] => [3,1,2,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[4,2,1,3,5] => [4,2,1,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[4,2,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[4,2,3,5,1] => [3,1,2,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[4,3,1,2,5] => [4,3,1,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[4,5,1,2,3] => [3,5,1,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[5,1,2,3,4] => [5,1,2,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[5,1,3,2,4] => [5,1,3,2,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000668
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,1}
[2,1] => [1,2] => [2]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[1,3,2] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,1,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,3,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,1,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[3,2,1] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,3,2,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,3,4,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,4,2,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,4,3,2] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,3,1,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,4,1,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,1,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,4,1] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,4,1,2] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,4,2,1] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,1,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,2,1,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,1,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,2,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,5,2] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,2,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,1,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,4,5,1,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,4,5,2,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,5,1,2,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,5,1,4,2] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,5,2,1,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,5,2,4,1] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,1,2,5,3] => [1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 1
[4,1,3,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,1,5,3,2] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,2,1,5,3] => [1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 1
[4,2,3,5,1] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,2,5,3,1] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[4,3,1,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,3,2,5,1] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,5,1,2,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,5,1,3,2] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,5,2,1,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,5,2,3,1] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,5,3,1,2] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,5,3,2,1] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[5,1,2,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[5,1,2,4,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,1,3,2,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,1,4,2,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[5,2,1,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[5,2,1,4,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,2,3,1,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,2,4,1,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[5,3,1,2,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,3,2,1,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,1,2,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,4,1,3,2] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,2,1,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,4,2,3,1] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,3,1,2] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,3,2,1] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000707
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,1}
[2,1] => [1,2] => [2]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[1,3,2] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,1,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,3,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,1,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[3,2,1] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,3,2,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,3,4,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,4,2,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,4,3,2] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,3,1,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,4,1,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,1,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,4,1] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,4,1,2] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,4,2,1] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,1,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,2,1,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,1,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,2,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,5,2] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,2,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,1,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,4,5,1,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,4,5,2,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,5,1,2,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,5,1,4,2] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,5,2,1,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,5,2,4,1] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,1,2,5,3] => [1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 1
[4,1,3,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,1,5,3,2] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,2,1,5,3] => [1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 1
[4,2,3,5,1] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,2,5,3,1] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[4,3,1,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,3,2,5,1] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,5,1,2,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,5,1,3,2] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,5,2,1,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,5,2,3,1] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,5,3,1,2] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,5,3,2,1] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[5,1,2,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[5,1,2,4,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,1,3,2,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,1,4,2,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[5,2,1,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[5,2,1,4,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,2,3,1,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,2,4,1,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[5,3,1,2,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,3,2,1,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,1,2,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,4,1,3,2] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,2,1,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,4,2,3,1] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,3,1,2] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,3,2,1] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 1
Description
The product of the factorials of the parts.
Matching statistic: St000708
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,1}
[2,1] => [1,2] => [2]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[1,3,2] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,1,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,3,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,1,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[3,2,1] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,3,2,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,3,4,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,4,2,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,4,3,2] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,3,1,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,4,1,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,1,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,4,1] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,4,1,2] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,4,2,1] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,1,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,2,1,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,1,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,2,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,5,2] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,2,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,1,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,4,5,1,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,4,5,2,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,5,1,2,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,5,1,4,2] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,5,2,1,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,5,2,4,1] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,1,2,5,3] => [1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 1
[4,1,3,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,1,5,3,2] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,2,1,5,3] => [1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 1
[4,2,3,5,1] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,2,5,3,1] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[4,3,1,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,3,2,5,1] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,5,1,2,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,5,1,3,2] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,5,2,1,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,5,2,3,1] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,5,3,1,2] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,5,3,2,1] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[5,1,2,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[5,1,2,4,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,1,3,2,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,1,4,2,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[5,2,1,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[5,2,1,4,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,2,3,1,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,2,4,1,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[5,3,1,2,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,3,2,1,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,1,2,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,4,1,3,2] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,2,1,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,4,2,3,1] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,3,1,2] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,3,2,1] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 1
Description
The product of the parts of an integer partition.
Matching statistic: St000933
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,1}
[2,1] => [1,2] => [2]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[1,3,2] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,1,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,3,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,1,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[3,2,1] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,3,2,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,3,4,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,4,2,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,4,3,2] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,3,1,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,4,1,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,1,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,4,1] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,4,1,2] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,4,2,1] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,1,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,2,1,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,1,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,2,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,5,2] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,2,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,1,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,4,5,1,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,4,5,2,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,5,1,2,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,5,1,4,2] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,5,2,1,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,5,2,4,1] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,1,2,5,3] => [1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 1
[4,1,3,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,1,5,3,2] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,2,1,5,3] => [1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 1
[4,2,3,5,1] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,2,5,3,1] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[4,3,1,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,3,2,5,1] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,5,1,2,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,5,1,3,2] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,5,2,1,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,5,2,3,1] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,5,3,1,2] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,5,3,2,1] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[5,1,2,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[5,1,2,4,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,1,3,2,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,1,4,2,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[5,2,1,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[5,2,1,4,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,2,3,1,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,2,4,1,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[5,3,1,2,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,3,2,1,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,1,2,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,4,1,3,2] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,2,1,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,4,2,3,1] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,3,1,2] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,3,2,1] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 1
Description
The number of multipartitions of sizes given by an integer partition.
This is, for λ=(λ1,…,λn), this is the number of n-tuples (λ(1),…,λ(n)) of partitions λ(i) such that λ(i)⊢λi.
Matching statistic: St001568
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,1}
[2,1] => [1,2] => [2]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[1,3,2] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,1,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,3,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,1,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[3,2,1] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1}
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,2,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,4,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,2,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,3,2] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,1,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,1,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,1,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,4,1] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,1,2] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,2,1] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 2
[4,1,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2,1,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 2
[4,2,3,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,1,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,2,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,4,5,3] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,5,3,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,5,4,3] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,5,2] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,5,2,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,5,4,2] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,5,2] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,5,2,3] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,5,3,2] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2
[1,5,2,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2
[1,5,3,4,2] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2
[3,1,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[3,2,5,1,4] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2
[3,2,5,4,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[3,4,5,1,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[3,4,5,2,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[3,5,1,2,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[3,5,1,4,2] => [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[3,5,2,1,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[3,5,2,4,1] => [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2
[4,1,2,5,3] => [1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 2
[4,1,3,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[4,1,5,3,2] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 2
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2
[4,2,1,5,3] => [1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 2
[4,2,3,5,1] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[4,2,5,3,1] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 2
[4,3,1,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[4,3,2,5,1] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[4,5,1,2,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[4,5,1,3,2] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2
[4,5,2,1,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[4,5,2,3,1] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2
[4,5,3,1,2] => [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[4,5,3,2,1] => [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[5,1,2,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 2
[5,1,2,4,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 2
[5,1,3,2,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 2
[5,1,4,2,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 2
[5,2,1,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 2
[5,2,1,4,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 2
[5,2,3,1,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 2
[5,2,4,1,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 2
[5,3,1,2,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 2
[5,3,2,1,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 2
[5,4,1,2,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 2
[5,4,1,3,2] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 2
[5,4,2,1,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 2
[5,4,2,3,1] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 2
[5,4,3,1,2] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 2
[5,4,3,2,1] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 2
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 2
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 2
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000510
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 75%
Values
[1,2] => [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,1}
[1,2,3] => [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1}
[2,1,3] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[3,2,1] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[1,4,2,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,1]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[3,1,2,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[3,4,1,2] => [2,2]
=> [2]
=> [1,1]
=> 1
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[4,3,2,1] => [2,2]
=> [2]
=> [1,1]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[1,4,5,3,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,3,4,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,1,4,2,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,4,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,2,4,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,5,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,4,2,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,5,2,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,4,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The number of invariant oriented cycles when acting with a permutation of given cycle type.
Matching statistic: St000681
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,1}
[1,2,3] => [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1}
[2,1,3] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[3,2,1] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[1,4,2,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,1]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[3,1,2,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[3,4,1,2] => [2,2]
=> [2]
=> [1,1]
=> 1
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2}
[4,3,2,1] => [2,2]
=> [2]
=> [1,1]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[1,4,5,3,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,3,4,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,1,4,2,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,4,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,2,4,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,5,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,4,2,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,5,2,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,4,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
The following 74 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St000456The monochromatic index of a connected graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000937The number of positive values of the symmetric group character corresponding to the partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001280The number of parts of an integer partition that are at least two. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St000137The Grundy value of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000284The Plancherel distribution on integer partitions. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001128The exponens consonantiae of a partition. St000741The Colin de Verdière graph invariant. St001877Number of indecomposable injective modules with projective dimension 2. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000934The 2-degree of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000454The largest eigenvalue of a graph if it is integral. St000259The diameter of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000706The product of the factorials of the multiplicities of an integer partition. St000944The 3-degree of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001896The number of right descents of a signed permutations. St001427The number of descents of a signed permutation. St001621The number of atoms of a lattice.
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