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Your data matches 59 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St000282
Values
([],2)
=> 0
([(0,1)],2)
=> 1
([],3)
=> 0
([(1,2)],3)
=> 0
([(0,1),(0,2)],3)
=> 0
([(0,2),(2,1)],3)
=> 1
([(0,2),(1,2)],3)
=> 1
([],4)
=> 0
([(2,3)],4)
=> 0
([(1,2),(1,3)],4)
=> 0
([(0,1),(0,2),(0,3)],4)
=> 0
([(0,2),(0,3),(3,1)],4)
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(1,2),(2,3)],4)
=> 0
([(0,3),(3,1),(3,2)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(3,2)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(1,3)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
([(0,3),(2,1),(3,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> 2
([],5)
=> 0
([(3,4)],5)
=> 0
([(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(1,4)],5)
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 0
([(1,3),(1,4),(4,2)],5)
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 0
([(2,3),(3,4)],5)
=> 0
([(1,4),(4,2),(4,3)],5)
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(4,3)],5)
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0
Description
The size of the preimage of the map 'to poset' from Ordered trees to Posets.
Matching statistic: St001570
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1}
([(0,1)],2)
=> ([],2)
=> ([],2)
=> ([],1)
=> ? ∊ {0,1}
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1}
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1}
([(0,2),(2,1)],3)
=> ([],3)
=> ([],3)
=> ([],1)
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1}
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,1,1,1,2}
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,1,1,1,2}
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,1,1,1,2}
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,1,1,1,2}
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,1,1,1,2}
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,1,1,1,2}
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St000175
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Values
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
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 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Values
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
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 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Values
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
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]].
Matching statistic: St000225
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Values
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000749
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Values
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
Description
The 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.
For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields
$$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3.
This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
Matching statistic: St000944
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000944: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000944: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Values
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
Description
The 3-degree of an integer partition.
For an integer partition $\lambda$, this is given by the exponent of 3 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
This stupid comment should not be accepted as an edit!
Matching statistic: St001175
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Values
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St001178
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001178: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001178: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 71%●distinct values known / distinct values provided: 20%
Values
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4}
Description
Twelve times the variance of the major index among all standard Young tableaux of a partition.
For a partition $\lambda$ of $n$, this variance is given in [1, Proposition 3.2] by
$$\frac{1}{12}\Big(\sum_{k = 1}^n i^2 - \sum_{i,j \in \lambda} h_{ij}^2\Big),$$
where the second sum ranges over all cells in $\lambda$ and $h_{ij}$ is the hook length of the cell $(i,j) \in \lambda$.
The following 49 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001586The number of odd parts smaller than the largest even part in an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000929The constant term of the character polynomial of an integer partition. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001651The Frankl number of a lattice. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001964The interval resolution global dimension of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. 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. St001525The number of symmetric hooks on the diagonal of a partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. 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. St000655The length of the minimal rise of a Dyck path. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. 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. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001498The normalised height of a Nakayama algebra with magnitude 1. St000658The number of rises of length 2 of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. 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. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.
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