- FindStat

Your data matches 31 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000259
(load all 4 compositions to match this statistic)

Mapping

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

Matching statistic: St000667

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],1)
 => ([],1)
 => [1]
 => []
 => ? = 0
([],2)
 => ([],1)
 => [1]
 => []
 => ? = 0
([],3)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([],4)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([],5)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 2
([(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)
 => [4,1]
 => [1]
 => 1
([],6)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1
([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [6]
 => []
 => ? = 2
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([],7)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2

Description

The greatest common divisor of the parts of the partition.

Matching statistic: St001527

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001527: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],1)
 => ([],1)
 => [1]
 => []
 => ? = 0
([],2)
 => ([],1)
 => [1]
 => []
 => ? = 0
([],3)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([],4)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([],5)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 2
([(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)
 => [4,1]
 => [1]
 => 1
([],6)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1
([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [6]
 => []
 => ? = 2
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([],7)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2

Description

The cyclic permutation representation number of an integer partition. This is the size of the largest cyclic group

$C$ such that the fake degree is the character of a permutation representation of

$C$ .

Matching statistic: St001571

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],1)
 => ([],1)
 => [1]
 => []
 => ? = 0
([],2)
 => ([],1)
 => [1]
 => []
 => ? = 0
([],3)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([],4)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([],5)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 2
([(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)
 => [4,1]
 => [1]
 => 1
([],6)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1
([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [6]
 => []
 => ? = 2
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([],7)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2

Description

The Cartan determinant of the integer partition. Let

$p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let

$A=K[x]/(x^t)$ be the finite dimensional algebra over the field

$K$ and

$M$ the direct sum of the indecomposable

$A$ -modules of vector space dimension

$p_i$ for each

$i$ . Then the Cartan determinant of

$p$ is the Cartan determinant of the endomorphism algebra of

$M$ over

$A$ . Explicitly, this is the determinant of the matrix

$\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$ , where

$\bar p$ is the set of distinct parts of the partition.

Matching statistic: St001606

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001606: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],1)
 => ([],1)
 => [1]
 => []
 => ? = 0
([],2)
 => ([],1)
 => [1]
 => []
 => ? = 0
([],3)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([],4)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([],5)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 2
([(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)
 => [4,1]
 => [1]
 => 1
([],6)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1
([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [6]
 => []
 => ? = 2
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([],7)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2

Description

The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions.

Matching statistic: St001914

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001914: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],1)
 => ([],1)
 => [1]
 => []
 => ? = 0
([],2)
 => ([],1)
 => [1]
 => []
 => ? = 0
([],3)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([],4)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([],5)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 2
([(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)
 => [4,1]
 => [1]
 => 1
([],6)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1
([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [6]
 => []
 => ? = 2
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2
([],7)
 => ([],1)
 => [1]
 => []
 => ? = 0
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2

Description

The size of the orbit of an integer partition in Bulgarian solitaire. Bulgarian solitaire is the dynamical system where a move consists of removing the first column of the Ferrers diagram and inserting it as a row. This statistic returns the number of partitions that can be obtained from the given partition.

Matching statistic: St000145

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],1)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([],2)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([],3)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([],4)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([],5)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 2 - 1
([(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)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([],6)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 1 - 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([],7)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1

Description

The Dyson rank of a partition. This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences

$p(5n+4) \equiv 0 \pmod 5$ and

$p(7n+6) \equiv 0 \pmod 7.$

Matching statistic: St001384

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],1)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([],2)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([],3)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([],4)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([],5)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 2 - 1
([(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)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([],6)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 1 - 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([],7)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1

Description

The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.

Matching statistic: St001392

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],1)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([],2)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([],3)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([],4)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([],5)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 2 - 1
([(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)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([],6)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 1 - 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([],7)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1

Description

The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.

Matching statistic: St001767

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001767: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],1)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([],2)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([],3)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([],4)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([],5)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 2 - 1
([(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)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([],6)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 1 - 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 - 1
([],7)
 => ([],1)
 => [1]
 => []
 => ? = 0 - 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 - 1

Description

The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. Assign to each cell of the Ferrers diagram an arrow pointing north, east, south or west. Then compute for each cell the number of arrows pointing towards it, and take the minimum of those. This statistic is the maximal minimum that can be obtained by assigning arrows in any way.

The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000455The second largest eigenvalue of a graph if it is integral. St000456The monochromatic index of a connected graph. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000934The 2-degree of an integer partition. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001625The Möbius invariant of a lattice. St001626The number of maximal proper sublattices of a lattice. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001877Number of indecomposable injective modules with projective dimension 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. St001875The number of simple modules with projective dimension at most 1. St000454The largest eigenvalue of a graph if it is integral.