- FindStat

Your data matches 102 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001073
(load all 2 compositions to match this statistic)

Mapping

St001073: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 1
([],2)
 => 1
([(0,1)],2)
 => 0
([],3)
 => 1
([(1,2)],3)
 => 0
([(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(1,2)],3)
 => 2
([],4)
 => 1
([(2,3)],4)
 => 0
([(1,3),(2,3)],4)
 => 0
([(0,3),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2)],4)
 => 0
([(0,3),(1,2),(2,3)],4)
 => 0
([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([],5)
 => 1
([(3,4)],5)
 => 0
([(2,4),(3,4)],5)
 => 0
([(1,4),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
([(1,4),(2,3)],5)
 => 0
([(1,4),(2,3),(3,4)],5)
 => 0
([(0,1),(2,4),(3,4)],5)
 => 0
([(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2

Description

The number of nowhere zero 3-flows of a graph. This is the absolute value of the evaluation of the Tutte polynomial of the graph at $(x,y)=(0,-2)$. For $4$-regular graphs, this coincides with the number of Eulerian orientations.

Matching statistic: St001570
(load all 12 compositions to match this statistic)

Mapping

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 53%●distinct values known / distinct values provided: 6%

Values

([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 1
([],2)
 => ([],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {0,1}
([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1}
([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,2}
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,2}
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,1,2}
([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,2,2,2}
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,2,2,2}
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,2,2,2}
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,2,2,2}
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,0,1,2,2,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,0,1,2,2,2}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,0,1,2,2,2}
([],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(3,4)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(4,5)],6)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(2,5),(3,5),(4,5)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,2),(3,5),(4,5)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ([],1)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ([],1)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ([],1)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ([],1)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ([],1)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ([],1)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ([],1)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}

Description

The minimal number of edges to add to make a graph Hamiltonian. A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.

Matching statistic: St000205
(load all 10 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 52%●distinct values known / distinct values provided: 6%

Values

([],1)
 => []
 => ?
 => ? = 1
([],2)
 => []
 => ?
 => ? ∊ {0,1}
([(0,1)],2)
 => [1]
 => []
 => ? ∊ {0,1}
([],3)
 => []
 => ?
 => ? ∊ {0,1,2}
([(1,2)],3)
 => [1]
 => []
 => ? ∊ {0,1,2}
([(0,2),(1,2)],3)
 => [1,1]
 => [1]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,1,2}
([],4)
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2}
([(2,3)],4)
 => [1]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(1,3),(2,3)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2}
([],5)
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(3,4)],5)
 => [1]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(2,4),(3,4)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([],6)
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(4,5)],6)
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(3,5),(4,5)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1]
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}

Description

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.

Matching statistic: St000206
(load all 10 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 52%●distinct values known / distinct values provided: 6%

Values

([],1)
 => []
 => ?
 => ? = 1
([],2)
 => []
 => ?
 => ? ∊ {0,1}
([(0,1)],2)
 => [1]
 => []
 => ? ∊ {0,1}
([],3)
 => []
 => ?
 => ? ∊ {0,1,2}
([(1,2)],3)
 => [1]
 => []
 => ? ∊ {0,1,2}
([(0,2),(1,2)],3)
 => [1,1]
 => [1]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,1,2}
([],4)
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2}
([(2,3)],4)
 => [1]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(1,3),(2,3)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2}
([],5)
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(3,4)],5)
 => [1]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(2,4),(3,4)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([],6)
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(4,5)],6)
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(3,5),(4,5)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1]
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}

Description

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex. See also [[St000205]]. Each value in this statistic is greater than or equal to corresponding value in [[St000205]].

Matching statistic: St000225
(load all 2 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 52%●distinct values known / distinct values provided: 12%

Values

([],1)
 => []
 => ?
 => ? = 1
([],2)
 => []
 => ?
 => ? ∊ {0,1}
([(0,1)],2)
 => [1]
 => []
 => ? ∊ {0,1}
([],3)
 => []
 => ?
 => ? ∊ {0,1,2}
([(1,2)],3)
 => [1]
 => []
 => ? ∊ {0,1,2}
([(0,2),(1,2)],3)
 => [1,1]
 => [1]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,1,2}
([],4)
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2}
([(2,3)],4)
 => [1]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(1,3),(2,3)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2}
([],5)
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(3,4)],5)
 => [1]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(2,4),(3,4)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([],6)
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(4,5)],6)
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(3,5),(4,5)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1]
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}

Description

Difference between largest and smallest parts in a partition.

Matching statistic: St000319
(load all 2 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 52%●distinct values known / distinct values provided: 12%

Values

([],1)
 => []
 => ?
 => ? = 1
([],2)
 => []
 => ?
 => ? ∊ {0,1}
([(0,1)],2)
 => [1]
 => []
 => ? ∊ {0,1}
([],3)
 => []
 => ?
 => ? ∊ {0,1,2}
([(1,2)],3)
 => [1]
 => []
 => ? ∊ {0,1,2}
([(0,2),(1,2)],3)
 => [1,1]
 => [1]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,1,2}
([],4)
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2}
([(2,3)],4)
 => [1]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(1,3),(2,3)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2}
([],5)
 => []
 => ?
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(3,4)],5)
 => [1]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(2,4),(3,4)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([],6)
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(4,5)],6)
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(3,5),(4,5)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1]
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}

Description

The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.

Matching statistic: St000320
(load all 2 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 52%●distinct values known / distinct values provided: 12%

Values

([],1)
 => []
 => ?
 => ? = 1
([],2)
 => []
 => ?
 => ? ∊ {0,1}
([(0,1)],2)
 => [1]
 => []
 => ? ∊ {0,1}
([],3)
 => []
 => ?
 => ? ∊ {0,1,2}
([(1,2)],3)
 => [1]
 => []
 => ? ∊ {0,1,2}
([(0,2),(1,2)],3)
 => [1,1]
 => [1]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,1,2}
([],4)
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2}
([(2,3)],4)
 => [1]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(1,3),(2,3)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2}
([],5)
 => []
 => ?
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(3,4)],5)
 => [1]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(2,4),(3,4)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,4,4,4,6,24}
([],6)
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(4,5)],6)
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(3,5),(4,5)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1]
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}

Description

The dinv adjustment of an integer partition. The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$. The dinv adjustment is then defined by $$\sum_{j:n_j > 0}(\lambda_1-1-j).$$ The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$ and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$. The dinv adjustment is thus $4+3+1+0 = 8$.

Matching statistic: St000749
(load all 3 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 52%●distinct values known / distinct values provided: 6%

Values

([],1)
 => []
 => ?
 => ? = 1
([],2)
 => []
 => ?
 => ? ∊ {0,1}
([(0,1)],2)
 => [1]
 => []
 => ? ∊ {0,1}
([],3)
 => []
 => ?
 => ? ∊ {0,1,2}
([(1,2)],3)
 => [1]
 => []
 => ? ∊ {0,1,2}
([(0,2),(1,2)],3)
 => [1,1]
 => [1]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,1,2}
([],4)
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2}
([(2,3)],4)
 => [1]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(1,3),(2,3)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2}
([],5)
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(3,4)],5)
 => [1]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(2,4),(3,4)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([],6)
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(4,5)],6)
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(3,5),(4,5)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1]
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}

Description

The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields $$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3. This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].

Matching statistic: St001175
(load all 8 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 52%●distinct values known / distinct values provided: 6%

Values

([],1)
 => []
 => ?
 => ? = 1
([],2)
 => []
 => ?
 => ? ∊ {0,1}
([(0,1)],2)
 => [1]
 => []
 => ? ∊ {0,1}
([],3)
 => []
 => ?
 => ? ∊ {0,1,2}
([(1,2)],3)
 => [1]
 => []
 => ? ∊ {0,1,2}
([(0,2),(1,2)],3)
 => [1,1]
 => [1]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,1,2}
([],4)
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2}
([(2,3)],4)
 => [1]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(1,3),(2,3)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2}
([],5)
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(3,4)],5)
 => [1]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(2,4),(3,4)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([],6)
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(4,5)],6)
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(3,5),(4,5)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1]
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,8,8,8,10,10,12,12,16,20,24,24,24,32,38,44,72,80}

Description

The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.

Matching statistic: St001178
(load all 3 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001178: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 52%●distinct values known / distinct values provided: 12%

Values

([],1)
 => []
 => ?
 => ? = 1
([],2)
 => []
 => ?
 => ? ∊ {0,1}
([(0,1)],2)
 => [1]
 => []
 => ? ∊ {0,1}
([],3)
 => []
 => ?
 => ? ∊ {0,1,2}
([(1,2)],3)
 => [1]
 => []
 => ? ∊ {0,1,2}
([(0,2),(1,2)],3)
 => [1,1]
 => [1]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,1,2}
([],4)
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2}
([(2,3)],4)
 => [1]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(1,3),(2,3)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2}
([],5)
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(3,4)],5)
 => [1]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(2,4),(3,4)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => ? ∊ {0,0,1,2,2,2,2,2,2,2,2,4,4,4,6,24}
([],6)
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(4,5)],6)
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(3,5),(4,5)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1]
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,10,10,12,12,16,20,24,24,24,32,38,44,72,80}

Description

Twelve times the variance of the major index among all standard Young tableaux of a partition. For a partition $\lambda$ of $n$, this variance is given in [1, Proposition 3.2] by $$\frac{1}{12}\Big(\sum_{k = 1}^n i^2 - \sum_{i,j \in \lambda} h_{ij}^2\Big),$$ where the second sum ranges over all cells in $\lambda$ and $h_{ij}$ is the hook length of the cell $(i,j) \in \lambda$.

The following 92 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001248Sum of the even parts of a partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001479The number of bridges of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000017The number of inversions of a standard tableau. St000142The number of even parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000629The defect of a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000995The largest even part of an integer partition. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001092The number of distinct even parts of a partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001485The modular major index of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001596The number of two-by-two squares inside a skew partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001730The number of times the path corresponding to a binary word crosses the base line. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000940The number of characters of the symmetric group whose value on the partition is zero. 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. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St001651The Frankl number of a lattice. St001545The second Elser number of a connected graph. St000936The number of even values of the symmetric group character corresponding to the partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001060The distinguishing index of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000944The 3-degree of an integer partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001541The Gini index of an integer partition.