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Your data matches 171 different statistics following compositions of up to 3 maps.
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Matching statistic: St001070
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 2
([(0,1)],2)
=> 1
([],3)
=> 3
([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> 4
([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([],5)
=> 5
([(3,4)],5)
=> 1
([(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)
=> 0
([(1,4),(2,3),(3,4)],5)
=> 0
([(0,1),(2,4),(3,4)],5)
=> 0
([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
Description
The absolute value of the derivative of the chromatic polynomial of the graph at 1.
This is closely related to Crapo's beta invariant, the only difference being the value for the graphs without edges.
Matching statistic: St000929
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 13% ●values known / values provided: 65%●distinct values known / distinct values provided: 13%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 13% ●values known / values provided: 65%●distinct values known / distinct values provided: 13%
Values
([],1)
=> ([],1)
=> [1]
=> ? = 1
([],2)
=> ([],2)
=> [1,1]
=> 1
([(0,1)],2)
=> ([],1)
=> [1]
=> ? = 2
([],3)
=> ([],3)
=> [1,1,1]
=> 1
([(1,2)],3)
=> ([],2)
=> [1,1]
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> [1]
=> ? = 3
([],4)
=> ([],4)
=> [1,1,1,1]
=> 1
([(2,3)],4)
=> ([],3)
=> [1,1,1]
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> [1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> [1,1]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> [1]
=> ? ∊ {0,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1]
=> ? ∊ {0,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1]
=> ? ∊ {0,2,4}
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 1
([(3,4)],5)
=> ([],4)
=> [1,1,1,1]
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> [1,1,1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> [1,1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,2,2,2,3,4,5,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,2,2,2,3,4,5,6}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> [7]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,2,2,2,3,4,5,6}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,2,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,2,2,2,3,4,5,6}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,2,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,2,2,2,3,4,5,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,2,2,2,3,4,5,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,2,2,2,3,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,2,2,2,3,4,5,6}
([],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> 1
([(4,5)],6)
=> ([],5)
=> [1,1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> 0
([(2,5),(3,4)],6)
=> ([],4)
=> [1,1,1,1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> [1,1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [6,1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [7]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001392
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 53%●distinct values known / distinct values provided: 27%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 53%●distinct values known / distinct values provided: 27%
Values
([],1)
=> [1] => [[1],[]]
=> []
=> ? = 1
([],2)
=> [2] => [[2],[]]
=> []
=> ? ∊ {1,2}
([(0,1)],2)
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,2}
([],3)
=> [3] => [[3],[]]
=> []
=> ? ∊ {1,1,3}
([(1,2)],3)
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {1,1,3}
([(0,2),(1,2)],3)
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {1,1,3}
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [[2,2],[1]]
=> [1]
=> 0
([],4)
=> [4] => [[4],[]]
=> []
=> ? ∊ {1,1,1,2,4}
([(2,3)],4)
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {1,1,1,2,4}
([(1,3),(2,3)],4)
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,4}
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,4}
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
([],5)
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(3,4)],5)
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(2,4),(3,4)],5)
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
([(1,4),(2,3)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [[4,4],[3]]
=> [3]
=> 2
([],6)
=> [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(4,5)],6)
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(3,5),(4,5)],6)
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 2
([(2,5),(3,4)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 2
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> [2]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St001525
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001525: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 53%●distinct values known / distinct values provided: 20%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001525: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 53%●distinct values known / distinct values provided: 20%
Values
([],1)
=> [1] => [[1],[]]
=> []
=> ? = 1
([],2)
=> [2] => [[2],[]]
=> []
=> ? ∊ {1,2}
([(0,1)],2)
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,2}
([],3)
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,1,3}
([(1,2)],3)
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,3}
([(0,2),(1,2)],3)
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,3}
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [[2,2],[1]]
=> [1]
=> 1
([],4)
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,2,4}
([(2,3)],4)
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,2,4}
([(1,3),(2,3)],4)
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,2,4}
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,2,4}
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,2,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [[3,3],[2]]
=> [2]
=> 0
([],5)
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(3,4)],5)
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(2,4),(3,4)],5)
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
([(1,4),(2,3)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [[4,3],[2]]
=> [2]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0
([],6)
=> [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(4,5)],6)
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(3,5),(4,5)],6)
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0
([(2,5),(3,4)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> [2]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
Description
The number of symmetric hooks on the diagonal of a partition.
Matching statistic: St001657
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 53%●distinct values known / distinct values provided: 27%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 53%●distinct values known / distinct values provided: 27%
Values
([],1)
=> [1] => [[1],[]]
=> []
=> ? = 1
([],2)
=> [2] => [[2],[]]
=> []
=> ? ∊ {1,2}
([(0,1)],2)
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,2}
([],3)
=> [3] => [[3],[]]
=> []
=> ? ∊ {1,1,3}
([(1,2)],3)
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {1,1,3}
([(0,2),(1,2)],3)
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {1,1,3}
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [[2,2],[1]]
=> [1]
=> 0
([],4)
=> [4] => [[4],[]]
=> []
=> ? ∊ {1,1,1,2,4}
([(2,3)],4)
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {1,1,1,2,4}
([(1,3),(2,3)],4)
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,4}
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,4}
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
([],5)
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(3,4)],5)
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(2,4),(3,4)],5)
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
([(1,4),(2,3)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,4,5,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0
([],6)
=> [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(4,5)],6)
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(3,5),(4,5)],6)
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0
([(2,5),(3,4)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> [2]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St001939
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001939: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 53%●distinct values known / distinct values provided: 27%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001939: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 53%●distinct values known / distinct values provided: 27%
Values
([],1)
=> [1] => [[1],[]]
=> []
=> ? = 1
([],2)
=> [2] => [[2],[]]
=> []
=> ? ∊ {1,2}
([(0,1)],2)
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,2}
([],3)
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,1,3}
([(1,2)],3)
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,3}
([(0,2),(1,2)],3)
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,3}
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [[2,2],[1]]
=> [1]
=> 1
([],4)
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,2,4}
([(2,3)],4)
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,2,4}
([(1,3),(2,3)],4)
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,2,4}
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,2,4}
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [[3,2],[1]]
=> [1]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,2,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [[3,3],[2]]
=> [2]
=> 0
([],5)
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(3,4)],5)
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(2,4),(3,4)],5)
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
([(1,4),(2,3)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [[4,2],[1]]
=> [1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [[4,3],[2]]
=> [2]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,4,5,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0
([],6)
=> [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(4,5)],6)
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(3,5),(4,5)],6)
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0
([(2,5),(3,4)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [[5,2],[1]]
=> [1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> [2]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
Description
The number of parts that are equal to their multiplicity in the integer partition.
Matching statistic: St001629
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 27% ●values known / values provided: 53%●distinct values known / distinct values provided: 27%
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 27% ●values known / values provided: 53%●distinct values known / distinct values provided: 27%
Values
([],1)
=> ([],1)
=> [1] => ? = 1
([],2)
=> ([],2)
=> [2] => ? ∊ {1,2}
([(0,1)],2)
=> ([],1)
=> [1] => ? ∊ {1,2}
([],3)
=> ([],3)
=> [3] => 1
([(1,2)],3)
=> ([],2)
=> [2] => ? ∊ {1,3}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> [1] => ? ∊ {1,3}
([],4)
=> ([],4)
=> [4] => 1
([(2,3)],4)
=> ([],3)
=> [3] => 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1] => 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3,1] => 0
([(0,3),(1,2)],4)
=> ([],2)
=> [2] => ? ∊ {0,0,1,2,4}
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 1
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> [2] => ? ∊ {0,0,1,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> [1] => ? ∊ {0,0,1,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1] => ? ∊ {0,0,1,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> [1] => ? ∊ {0,0,1,2,4}
([],5)
=> ([],5)
=> [5] => 1
([(3,4)],5)
=> ([],4)
=> [4] => 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 0
([(1,4),(2,3)],5)
=> ([],3)
=> [3] => 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [4,2] => 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1] => 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> [3] => 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [4,2] => 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1] => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> [3,1] => 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> [2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,5,6}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,5,6}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,5,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,5,6}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> [4,3] => 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,5,6}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,5,6}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,5,6}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,5,6}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,5,6}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,5,6}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,5,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,5,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,5,6}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3,5,6}
([],6)
=> ([],6)
=> [6] => 1
([(4,5)],6)
=> ([],5)
=> [5] => 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [5,1] => 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [5,1] => 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => 0
([(2,5),(3,4)],6)
=> ([],4)
=> [4] => 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,2] => 2
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> [4,1] => 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> [4] => 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [5,2] => 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> [4,1] => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [5,2] => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([],3)
=> [3] => 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> [3,1] => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> [3] => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [4,2] => 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> [5,2] => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> [2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> [2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([],2)
=> [2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> [2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> [2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> [2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> [2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> [2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> [2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> [2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St000175
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 13% ●values known / values provided: 52%●distinct values known / distinct values provided: 13%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 13% ●values known / values provided: 52%●distinct values known / distinct values provided: 13%
Values
([],1)
=> []
=> ?
=> ? = 1
([],2)
=> []
=> ?
=> ? ∊ {1,2}
([(0,1)],2)
=> [1]
=> []
=> ? ∊ {1,2}
([],3)
=> []
=> ?
=> ? ∊ {1,1,3}
([(1,2)],3)
=> [1]
=> []
=> ? ∊ {1,1,3}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,1,3}
([],4)
=> []
=> ?
=> ? ∊ {1,1,1,1,2,4}
([(2,3)],4)
=> [1]
=> []
=> ? ∊ {1,1,1,1,2,4}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> ? ∊ {1,1,1,1,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> ? ∊ {1,1,1,1,2,4}
([],5)
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(3,4)],5)
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([],6)
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(4,5)],6)
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [8]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [9]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
Description
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape.
Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial
$$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$
The statistic of the degree of this polynomial.
For example, the partition $(3, 2, 1, 1, 1)$ gives
$$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$
which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$.
This is the same as the number of unordered pairs of different parts, which follows from:
$$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Matching statistic: St000205
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 7% ●values known / values provided: 52%●distinct values known / distinct values provided: 7%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 7% ●values known / values provided: 52%●distinct values known / distinct values provided: 7%
Values
([],1)
=> []
=> ?
=> ? = 1
([],2)
=> []
=> ?
=> ? ∊ {1,2}
([(0,1)],2)
=> [1]
=> []
=> ? ∊ {1,2}
([],3)
=> []
=> ?
=> ? ∊ {1,1,3}
([(1,2)],3)
=> [1]
=> []
=> ? ∊ {1,1,3}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,1,3}
([],4)
=> []
=> ?
=> ? ∊ {1,1,1,1,2,4}
([(2,3)],4)
=> [1]
=> []
=> ? ∊ {1,1,1,1,2,4}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> ? ∊ {1,1,1,1,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> ? ∊ {1,1,1,1,2,4}
([],5)
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(3,4)],5)
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([],6)
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(4,5)],6)
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [9]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 7% ●values known / values provided: 52%●distinct values known / distinct values provided: 7%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 7% ●values known / values provided: 52%●distinct values known / distinct values provided: 7%
Values
([],1)
=> []
=> ?
=> ? = 1
([],2)
=> []
=> ?
=> ? ∊ {1,2}
([(0,1)],2)
=> [1]
=> []
=> ? ∊ {1,2}
([],3)
=> []
=> ?
=> ? ∊ {1,1,3}
([(1,2)],3)
=> [1]
=> []
=> ? ∊ {1,1,3}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,1,3}
([],4)
=> []
=> ?
=> ? ∊ {1,1,1,1,2,4}
([(2,3)],4)
=> [1]
=> []
=> ? ∊ {1,1,1,1,2,4}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> ? ∊ {1,1,1,1,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,1,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> ? ∊ {1,1,1,1,2,4}
([],5)
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(3,4)],5)
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6}
([],6)
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(4,5)],6)
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [9]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [8]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,10,11,12,14,18,24}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
The following 161 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000379The number of Hamiltonian cycles in a graph. St000455The second largest eigenvalue of a graph if it is integral. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001651The Frankl number of a lattice. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000475The number of parts equal to 1 in a partition. St000644The number of graphs with given frequency partition. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001249Sum of the odd parts of a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000941The number of characters of the symmetric group whose value on the partition is even. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000145The Dyson rank of a partition. St001279The sum of the parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001541The Gini index of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001961The sum of the greatest common divisors of all pairs of parts. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000567The sum of the products of all pairs of parts. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000706The product of the factorials of the multiplicities of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000874The position of the last double rise in a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000939The number of characters of the symmetric group whose value on the partition is positive. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000984The number of boxes below precisely one peak. St000993The multiplicity of the largest part of an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001568The smallest positive integer that does not appear twice in the partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type.
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