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Your data matches 52 different statistics following compositions of up to 3 maps.
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Matching statistic: St001319
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Values
([],1)
=> 0
([],2)
=> 0
([(0,1)],2)
=> 0
([],3)
=> 0
([(1,2)],3)
=> 0
([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> 0
([(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> 0
([(3,4)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(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)
=> 1
([(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)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(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)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 6
Description
The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph.
A graph is a disjoint union of isolated vertices and a star if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ is an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001553
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Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 29% ●values known / values provided: 34%●distinct values known / distinct values provided: 29%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 29% ●values known / values provided: 34%●distinct values known / distinct values provided: 29%
Values
([],1)
=> []
=> []
=> ? = 0
([],2)
=> []
=> []
=> ? = 0
([(0,1)],2)
=> [1]
=> [1,0]
=> 0
([],3)
=> []
=> []
=> ? = 0
([(1,2)],3)
=> [1]
=> [1,0]
=> 0
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([],4)
=> []
=> []
=> ? ∊ {1,4}
([(2,3)],4)
=> [1]
=> [1,0]
=> 0
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,4}
([],5)
=> []
=> []
=> ? ∊ {1,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(3,4)],5)
=> [1]
=> [1,0]
=> 0
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,3,3,4,4,4,4,4,4,5,5,6,7,10}
([],6)
=> []
=> []
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(4,5)],6)
=> [1]
=> [1,0]
=> 0
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {0,1,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
Description
The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
The statistic returns zero in case that bimodule is the zero module.
Matching statistic: St001118
Values
([],1)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 0
([],2)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? ∊ {0,0}
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0}
([],3)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? ∊ {0,0,0,1}
([(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1}
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {0,0,0,1}
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,1}
([],4)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? ∊ {0,0,0,0,1,1,2,2,4}
([(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2,4}
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {0,0,0,0,1,1,2,2,4}
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,4}
([(0,3),(1,2)],4)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,1,1,2,2,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,1,1,2,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {0,0,0,0,1,1,2,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? ∊ {0,0,0,0,1,1,2,2,4}
([],5)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,3)],5)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([],8)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([],8)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ([],9)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([],10)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,7,10}
([],6)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,5),(3,4)],6)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)
=> 4
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> 4
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,4),(1,5),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 4
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 4
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 4
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> 4
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 5
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5),(5,6)],7)
=> 5
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St001232
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 24%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 24%
Values
([],1)
=> []
=> []
=> []
=> ? = 0
([],2)
=> []
=> []
=> []
=> ? = 0
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([],3)
=> []
=> []
=> []
=> ? ∊ {0,0}
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,2),(1,2)],3)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {0,0}
([],4)
=> []
=> []
=> []
=> ? ∊ {0,0,0,1,1,2,4}
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(1,3),(2,3)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,1,1,2,4}
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,1,1,2,4}
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,1,1,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,1,1,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,1,1,2,4}
([],5)
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(2,4),(3,4)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([],6)
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(3,5),(4,5)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000668
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 18%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 18%
Values
([],1)
=> [1]
=> []
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> ? ∊ {0,0}
([(0,1)],2)
=> [2]
=> []
=> ? ∊ {0,0}
([],3)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {0,0,0}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,0}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,0}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
Description
The least common multiple of the parts of the partition.
Matching statistic: St000707
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 24%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 24%
Values
([],1)
=> [1]
=> []
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> ? ∊ {0,0}
([(0,1)],2)
=> [2]
=> []
=> ? ∊ {0,0}
([],3)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {0,0,0}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,0}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,0}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 6
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 4
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 6
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 6
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
Description
The product of the factorials of the parts.
Matching statistic: St000708
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 24%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 24%
Values
([],1)
=> [1]
=> []
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> ? ∊ {0,0}
([(0,1)],2)
=> [2]
=> []
=> ? ∊ {0,0}
([],3)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {0,0,0}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,0}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,0}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 4
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
Description
The product of the parts of an integer partition.
Matching statistic: St000770
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 29%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 29%
Values
([],1)
=> [1]
=> []
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> ? ∊ {0,0}
([(0,1)],2)
=> [2]
=> []
=> ? ∊ {0,0}
([],3)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {0,0,0}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,0}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,0}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 4
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,6,7,10}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 5
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 4
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 4
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
Description
The major index of an integer partition when read from bottom to top.
This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top.
For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
Matching statistic: St000815
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000815: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 24%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000815: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 24%
Values
([],1)
=> [1]
=> []
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> ? ∊ {0,0}
([(0,1)],2)
=> [2]
=> []
=> ? ∊ {0,0}
([],3)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {0,0,0}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,0}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,0}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 4
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 3
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 4
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
Description
The number of semistandard Young tableaux of partition weight of given shape.
The weight of a semistandard Young tableaux is the sequence $(m_1, m_2,\dots)$, where $m_i$ is the number of occurrences of the number $i$ in the tableau. This statistic counts those tableaux whose weight is a weakly decreasing sequence.
Alternatively, this is the sum of the entries in the column specified by the partition of the change of basis matrix from Schur functions to monomial symmetric functions.
Matching statistic: St000933
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 24%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 24%
Values
([],1)
=> [1]
=> []
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> ? ∊ {0,0}
([(0,1)],2)
=> [2]
=> []
=> ? ∊ {0,0}
([],3)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {0,0,0}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,0}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,0}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,7,10}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,13,14,14,16,20}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 4
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
Description
The number of multipartitions of sizes given by an integer partition.
This is, for $\lambda = (\lambda_1,\ldots,\lambda_n)$, this is the number of $n$-tuples $(\lambda^{(1)},\ldots,\lambda^{(n)})$ of partitions $\lambda^{(i)}$ such that $\lambda^{(i)} \vdash \lambda_i$.
The following 42 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000937The number of positive values of the symmetric group character corresponding to the partition. St000420The number of Dyck paths that are weakly above a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000939The number of characters of the symmetric group whose value on the partition is positive. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000456The monochromatic index of a connected graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St001626The number of maximal proper sublattices of a lattice. 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 $eA$. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to 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. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an 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. St001593This is the number of standard Young tableaux of the given shifted shape. 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. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. 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. St000264The girth of a graph, which is not a tree. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St000273The domination number of a graph. St001829The common independence number of a graph.
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