searching the database
Your data matches 26 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001635
Values
([],1)
=> 1
([],2)
=> 2
([(0,1)],2)
=> -1
([],3)
=> 3
([(1,2)],3)
=> 0
([(0,1),(0,2)],3)
=> -1
([(0,2),(2,1)],3)
=> -1
([(0,2),(1,2)],3)
=> -1
([],4)
=> 4
([(2,3)],4)
=> 1
([(1,2),(1,3)],4)
=> 0
([(0,1),(0,2),(0,3)],4)
=> 1
([(0,2),(0,3),(3,1)],4)
=> -1
([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(1,2),(2,3)],4)
=> 0
([(0,3),(3,1),(3,2)],4)
=> 1
([(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)
=> -2
([(0,3),(1,2),(1,3)],4)
=> -1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(0,3),(2,1),(3,2)],4)
=> -1
([(0,3),(1,2),(2,3)],4)
=> -1
([],5)
=> 5
([(3,4)],5)
=> 2
([(2,3),(2,4)],5)
=> 1
([(1,2),(1,3),(1,4)],5)
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> 5
([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 5
([(1,3),(1,4),(4,2)],5)
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> -1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 5
([(2,3),(3,4)],5)
=> 1
([(1,4),(4,2),(4,3)],5)
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> 5
([(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(4,3)],5)
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> 5
([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> 5
([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
([(0,4),(1,4),(2,3)],5)
=> -2
([(0,4),(1,3),(2,3),(2,4)],5)
=> -1
Description
The trace of the square of the Coxeter matrix of the incidence algebra of a poset.
Matching statistic: St001118
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1
([],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {-1,2}
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-1,2}
([],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {-1,-1,-1,0,3}
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-1,-1,-1,0,3}
([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-1,-1,-1,0,3}
([],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(1,4),(1,5),(4,3),(5,2)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(1,5),(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(0,5),(1,5),(2,3),(5,4)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
([(0,5),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,4),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
([(0,5),(1,5),(2,3),(3,4)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(0,5),(1,4),(1,5),(4,2),(4,3)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
([(1,5),(2,3),(2,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,5),(1,3),(1,4),(1,5),(4,2)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(0,5),(1,2),(1,3),(1,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,4),(1,2),(1,3),(1,5),(4,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
([(0,3),(1,2),(1,4),(1,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St000284
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 5% ●values known / values provided: 24%●distinct values known / distinct values provided: 5%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 5% ●values known / values provided: 24%●distinct values known / distinct values provided: 5%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {-1,2}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {-1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-1,-1,-1,0,3}
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The Plancherel distribution on integer partitions.
This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions.
Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.
Matching statistic: St000510
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 24%●distinct values known / distinct values provided: 14%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 24%●distinct values known / distinct values provided: 14%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {-1,2}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {-1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-1,-1,-1,0,3}
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 6
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(0,5),(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The number of invariant oriented cycles when acting with a permutation of given cycle type.
Matching statistic: St000620
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 24%●distinct values known / distinct values provided: 9%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 24%●distinct values known / distinct values provided: 9%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {-1,2}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {-1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-1,-1,-1,0,3}
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd.
The case of an even minimum is [[St000621]].
Matching statistic: St000668
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 24%●distinct values known / distinct values provided: 9%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 24%●distinct values known / distinct values provided: 9%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {-1,2}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {-1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-1,-1,-1,0,3}
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000681
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 24%●distinct values known / distinct values provided: 14%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 24%●distinct values known / distinct values provided: 14%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {-1,2}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {-1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-1,-1,-1,0,3}
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(0,5),(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St000698
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 24%●distinct values known / distinct values provided: 9%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 24%●distinct values known / distinct values provided: 9%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {-1,2}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {-1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-1,-1,-1,0,3}
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St000704
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 5% ●values known / values provided: 24%●distinct values known / distinct values provided: 5%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 5% ●values known / values provided: 24%●distinct values known / distinct values provided: 5%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {-1,2}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {-1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-1,-1,-1,0,3}
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
Matching statistic: St000707
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 24%●distinct values known / distinct values provided: 9%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 24%●distinct values known / distinct values provided: 9%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {-1,2}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {-1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-1,-1,-1,0,3}
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {-1,-1,-1,0,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {-1,-1,-1,0,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,0,0,0,1,1,1,1,1,4,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,5,5,5,5,5,5,5,5,5,6,6,6,6,17,17}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The product of the factorials of the parts.
The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!