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Your data matches 220 different statistics following compositions of up to 3 maps.
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Matching statistic: St000986
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Values
([],1)
=> 1
([],2)
=> 2
([(0,1)],2)
=> 0
([],3)
=> 3
([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> 0
([],4)
=> 4
([(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([],5)
=> 5
([(3,4)],5)
=> 3
([(2,4),(3,4)],5)
=> 3
([(1,4),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3)],5)
=> 1
([(1,4),(2,3),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> 1
([(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
Matching statistic: St001384
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(load all 10 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 57%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 57%
Values
([],1)
=> [1] => [[1],[]]
=> []
=> ? = 1
([],2)
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,2}
([(0,1)],2)
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,2}
([],3)
=> [3] => [[3],[]]
=> []
=> ? ∊ {1,1,3}
([(1,2)],3)
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {1,1,3}
([(0,2),(1,2)],3)
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {1,1,3}
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [[2,2],[1]]
=> [1]
=> 0
([],4)
=> [4] => [[4],[]]
=> []
=> ? ∊ {2,2,2,2,4}
([(2,3)],4)
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {2,2,2,2,4}
([(1,3),(2,3)],4)
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {2,2,2,2,4}
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {2,2,2,2,4}
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {2,2,2,2,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
([],5)
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(3,4)],5)
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(2,4),(3,4)],5)
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
([(1,4),(2,3)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [[4,4],[3]]
=> [3]
=> 2
([],6)
=> [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(4,5)],6)
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(3,5),(4,5)],6)
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 2
([(2,5),(3,4)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> [2]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
Description
The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.
Matching statistic: St001176
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Values
([],1)
=> []
=> ?
=> ? = 1
([],2)
=> []
=> ?
=> ? ∊ {0,2}
([(0,1)],2)
=> [1]
=> []
=> ? ∊ {0,2}
([],3)
=> []
=> ?
=> ? ∊ {1,1,3}
([(1,2)],3)
=> [1]
=> []
=> ? ∊ {1,1,3}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,1,3}
([],4)
=> []
=> ?
=> ? ∊ {0,2,2,2,2,4}
([(2,3)],4)
=> [1]
=> []
=> ? ∊ {0,2,2,2,2,4}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> ? ∊ {0,2,2,2,2,4}
([],5)
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(3,4)],5)
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([],6)
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(4,5)],6)
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [8]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [9]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001714
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Values
([],1)
=> []
=> ?
=> ? = 1
([],2)
=> []
=> ?
=> ? ∊ {0,2}
([(0,1)],2)
=> [1]
=> []
=> ? ∊ {0,2}
([],3)
=> []
=> ?
=> ? ∊ {1,1,3}
([(1,2)],3)
=> [1]
=> []
=> ? ∊ {1,1,3}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,1,3}
([],4)
=> []
=> ?
=> ? ∊ {0,2,2,2,2,4}
([(2,3)],4)
=> [1]
=> []
=> ? ∊ {0,2,2,2,2,4}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> ? ∊ {0,2,2,2,2,4}
([],5)
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(3,4)],5)
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([],6)
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(4,5)],6)
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [8]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [9]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.
In particular, partitions with statistic $0$ are wide partitions.
Matching statistic: St000319
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Values
([],1)
=> []
=> ?
=> ?
=> ? = 1
([],2)
=> []
=> ?
=> ?
=> ? ∊ {0,2}
([(0,1)],2)
=> [1]
=> []
=> []
=> ? ∊ {0,2}
([],3)
=> []
=> ?
=> ?
=> ? ∊ {1,1,3}
([(1,2)],3)
=> [1]
=> []
=> []
=> ? ∊ {1,1,3}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> []
=> ? ∊ {1,1,3}
([],4)
=> []
=> ?
=> ?
=> ? ∊ {0,2,2,2,2,4}
([(2,3)],4)
=> [1]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([],5)
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(3,4)],5)
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> [1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([],6)
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(4,5)],6)
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> [1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1]
=> [1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1]
=> [1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [9]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Values
([],1)
=> []
=> ?
=> ?
=> ? = 1
([],2)
=> []
=> ?
=> ?
=> ? ∊ {0,2}
([(0,1)],2)
=> [1]
=> []
=> []
=> ? ∊ {0,2}
([],3)
=> []
=> ?
=> ?
=> ? ∊ {1,1,3}
([(1,2)],3)
=> [1]
=> []
=> []
=> ? ∊ {1,1,3}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> []
=> ? ∊ {1,1,3}
([],4)
=> []
=> ?
=> ?
=> ? ∊ {0,2,2,2,2,4}
([(2,3)],4)
=> [1]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([],5)
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(3,4)],5)
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> [1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([],6)
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(4,5)],6)
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> [1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1]
=> [1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1]
=> [1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [9]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001392
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Values
([],1)
=> []
=> ?
=> ?
=> ? = 1
([],2)
=> []
=> ?
=> ?
=> ? ∊ {0,2}
([(0,1)],2)
=> [1]
=> []
=> []
=> ? ∊ {0,2}
([],3)
=> []
=> ?
=> ?
=> ? ∊ {1,1,3}
([(1,2)],3)
=> [1]
=> []
=> []
=> ? ∊ {1,1,3}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> []
=> ? ∊ {1,1,3}
([],4)
=> []
=> ?
=> ?
=> ? ∊ {0,2,2,2,2,4}
([(2,3)],4)
=> [1]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([],5)
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(3,4)],5)
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> [1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([],6)
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(4,5)],6)
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> [1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1]
=> [1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1]
=> [1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [9]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St001918
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Values
([],1)
=> []
=> ?
=> ?
=> ? = 1
([],2)
=> []
=> ?
=> ?
=> ? ∊ {0,2}
([(0,1)],2)
=> [1]
=> []
=> []
=> ? ∊ {0,2}
([],3)
=> []
=> ?
=> ?
=> ? ∊ {1,1,3}
([(1,2)],3)
=> [1]
=> []
=> []
=> ? ∊ {1,1,3}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> []
=> ? ∊ {1,1,3}
([],4)
=> []
=> ?
=> ?
=> ? ∊ {0,2,2,2,2,4}
([(2,3)],4)
=> [1]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> []
=> ? ∊ {0,2,2,2,2,4}
([],5)
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(3,4)],5)
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> [1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,5}
([],6)
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(4,5)],6)
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> [1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1]
=> [1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1]
=> [1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [9]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,6}
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$.
The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is
$$
\sum_{p\in\lambda} [p]_{q^{N/p}},
$$
where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer.
This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals
$$
\left(1 - \frac{1}{\lambda_1}\right) N,
$$
where $\lambda_1$ is the largest part of $\lambda$.
The statistic is undefined for the empty partition.
Matching statistic: St000512
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 57%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 57%
Values
([],1)
=> [1] => [[1],[]]
=> []
=> ? = 1
([],2)
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,2}
([(0,1)],2)
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,2}
([],3)
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,3}
([(1,2)],3)
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,3}
([(0,2),(1,2)],3)
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,3}
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,3}
([],4)
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,2,2,2,2,4}
([(2,3)],4)
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,1,1,2,2,2,2,4}
([(1,3),(2,3)],4)
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,2,2,2,2,4}
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,2,2,2,2,4}
([(0,3),(1,2)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,2,2,2,2,4}
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,2,2,2,2,4}
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,2,2,2,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,2,2,2,2,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,2,2,2,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [[3,3],[2]]
=> [2]
=> 0
([],5)
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(3,4)],5)
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(2,4),(3,4)],5)
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
([(1,4),(2,3)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [[4,3],[2]]
=> [2]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1
([],6)
=> [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
([(4,5)],6)
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
([(3,5),(4,5)],6)
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
([(2,5),(3,4)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> [2]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 0
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> [2]
=> 0
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
Description
The number of invariant subsets of size 3 when acting with a permutation of given cycle type.
Matching statistic: St000621
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 44%●distinct values known / distinct values provided: 43%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 44%●distinct values known / distinct values provided: 43%
Values
([],1)
=> [1] => [[1],[]]
=> []
=> ? = 1
([],2)
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,2}
([(0,1)],2)
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,2}
([],3)
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,3}
([(1,2)],3)
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,3}
([(0,2),(1,2)],3)
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,3}
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,3}
([],4)
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,4}
([(2,3)],4)
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,4}
([(1,3),(2,3)],4)
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,4}
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,2,4}
([(0,3),(1,2)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,2,4}
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,4}
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
([],5)
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(3,4)],5)
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(2,4),(3,4)],5)
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
([(1,4),(2,3)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0
([],6)
=> [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
([(4,5)],6)
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
([(3,5),(4,5)],6)
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0
([(2,5),(3,4)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> [2]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> [2]
=> 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is [[St000620]].
The following 210 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000929The constant term of the character polynomial of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000455The second largest eigenvalue of a graph if it is integral. St001651The Frankl number of a lattice. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000010The length of the partition. St000012The area of a Dyck path. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000150The floored half-sum of the multiplicities of a partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000183The side length of the Durfee square of an integer partition. St000228The size of a partition. St000257The number of distinct parts of a partition that occur at least twice. St000295The length of the border of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000442The maximal area to the right of an up step of a Dyck path. St000459The hook length of the base cell of a partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000475The number of parts equal to 1 in a partition. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000519The largest length of a factor maximising the subword complexity. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000658The number of rises of length 2 of a Dyck path. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000867The sum of the hook lengths in the first row of an integer partition. St000885The number of critical steps in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000921The number of internal inversions of a binary word. St000992The alternating sum of the parts of an integer partition. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001127The sum of the squares of the parts of a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001413Half the length of the longest even length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001524The degree of symmetry of a binary word. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001803The maximal overlap of the cylindrical tableau associated with a tableau. 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). St001955The number of natural descents for set-valued two row standard Young tableaux. St000478Another weight of a partition according to Alladi. St000941The number of characters of the symmetric group whose value on the partition is even. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001541The Gini index of an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001570The minimal number of edges to add to make a graph Hamiltonian. St001118The acyclic chromatic index of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000567The sum of the products of all pairs of parts. St000659The number of rises of length at least 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000744The length of the path to the largest entry in a standard Young tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000874The position of the last double rise in a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000984The number of boxes below precisely one peak. St000993The multiplicity of the largest part of an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001128The exponens consonantiae of a partition. St001141The number of occurrences of hills of size 3 in a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001568The smallest positive integer that does not appear twice in the partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001060The distinguishing index of a graph. 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 $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. 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 $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000225Difference between largest and smallest parts in a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000944The 3-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. 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. St001360The number of covering relations in Young's lattice below a partition. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. 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. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts.
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