- FindStat

Your data matches 234 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001496
(load all 3 compositions to match this statistic)

Mapping

Mp00259: Graphs —vertex addition⟶ Graphs
Mp00318: Graphs —dual on components⟶ Graphs
St001496: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of graphs with the same Laplacian spectrum as the given graph.

Matching statistic: St001256
(load all 14 compositions to match this statistic)

Mapping

Mp00318: Graphs —dual on components⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001256: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 47%●distinct values known / distinct values provided: 33%

Values

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

Description

Number of simple reflexive modules that are 2-stable reflexive. See Definition 3.1. in the reference for the definition of 2-stable reflexive.

Matching statistic: St001198
(load all 12 compositions to match this statistic)

Mapping

Mp00318: Graphs —dual on components⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 47%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St001206
(load all 12 compositions to match this statistic)

Mapping

Mp00318: Graphs —dual on components⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 47%●distinct values known / distinct values provided: 33%

Values

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

Description

The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ .

Matching statistic: St001009

Mapping

Mp00157: Graphs —connected complement⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001009: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 46%●distinct values known / distinct values provided: 33%

Values

([],1)
 => ([],1)
 => []
 => []
 => ? = 1
([],2)
 => ([],2)
 => []
 => []
 => ? = 1
([(0,1)],2)
 => ([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => 1
([],3)
 => ([],3)
 => []
 => []
 => ? = 1
([(1,2)],3)
 => ([(1,2)],3)
 => [1]
 => [1,0,1,0]
 => 1
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([],4)
 => ([],4)
 => []
 => []
 => ? = 1
([(2,3)],4)
 => ([(2,3)],4)
 => [1]
 => [1,0,1,0]
 => 1
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([],5)
 => ([],5)
 => []
 => []
 => ? = 1
([(3,4)],5)
 => ([(3,4)],5)
 => [1]
 => [1,0,1,0]
 => 1
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 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)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([],6)
 => ([],6)
 => []
 => []
 => ? = 1
([(4,5)],6)
 => ([(4,5)],6)
 => [1]
 => [1,0,1,0]
 => 1
([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(2,5),(3,5),(4,5)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
([(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(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)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(0,5),(1,4),(1,5),(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)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 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)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 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)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1

Description

Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001011
(load all 2 compositions to match this statistic)

Mapping

Mp00157: Graphs —connected complement⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 46%●distinct values known / distinct values provided: 33%

Values

([],1)
 => ([],1)
 => []
 => []
 => ? = 1
([],2)
 => ([],2)
 => []
 => []
 => ? = 1
([(0,1)],2)
 => ([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => 1
([],3)
 => ([],3)
 => []
 => []
 => ? = 1
([(1,2)],3)
 => ([(1,2)],3)
 => [1]
 => [1,0,1,0]
 => 1
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([],4)
 => ([],4)
 => []
 => []
 => ? = 1
([(2,3)],4)
 => ([(2,3)],4)
 => [1]
 => [1,0,1,0]
 => 1
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([],5)
 => ([],5)
 => []
 => []
 => ? = 1
([(3,4)],5)
 => ([(3,4)],5)
 => [1]
 => [1,0,1,0]
 => 1
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 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)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([],6)
 => ([],6)
 => []
 => []
 => ? = 1
([(4,5)],6)
 => ([(4,5)],6)
 => [1]
 => [1,0,1,0]
 => 1
([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(2,5),(3,5),(4,5)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
([(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(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)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(0,5),(1,4),(1,5),(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)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 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)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 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)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1

Description

Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001063
(load all 3 compositions to match this statistic)

Mapping

Mp00157: Graphs —connected complement⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001063: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 46%●distinct values known / distinct values provided: 33%

Values

([],1)
 => ([],1)
 => []
 => []
 => ? = 1
([],2)
 => ([],2)
 => []
 => []
 => ? = 1
([(0,1)],2)
 => ([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => 1
([],3)
 => ([],3)
 => []
 => []
 => ? = 1
([(1,2)],3)
 => ([(1,2)],3)
 => [1]
 => [1,0,1,0]
 => 1
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([],4)
 => ([],4)
 => []
 => []
 => ? = 1
([(2,3)],4)
 => ([(2,3)],4)
 => [1]
 => [1,0,1,0]
 => 1
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([],5)
 => ([],5)
 => []
 => []
 => ? = 1
([(3,4)],5)
 => ([(3,4)],5)
 => [1]
 => [1,0,1,0]
 => 1
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 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)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([],6)
 => ([],6)
 => []
 => []
 => ? = 1
([(4,5)],6)
 => ([(4,5)],6)
 => [1]
 => [1,0,1,0]
 => 1
([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(2,5),(3,5),(4,5)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
([(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(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)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(0,5),(1,4),(1,5),(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)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 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)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 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)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1

Description

Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra.

Matching statistic: St001064
(load all 3 compositions to match this statistic)

Mapping

Mp00157: Graphs —connected complement⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001064: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 46%●distinct values known / distinct values provided: 33%

Values

([],1)
 => ([],1)
 => []
 => []
 => ? = 1
([],2)
 => ([],2)
 => []
 => []
 => ? = 1
([(0,1)],2)
 => ([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => 1
([],3)
 => ([],3)
 => []
 => []
 => ? = 1
([(1,2)],3)
 => ([(1,2)],3)
 => [1]
 => [1,0,1,0]
 => 1
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([],4)
 => ([],4)
 => []
 => []
 => ? = 1
([(2,3)],4)
 => ([(2,3)],4)
 => [1]
 => [1,0,1,0]
 => 1
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([],5)
 => ([],5)
 => []
 => []
 => ? = 1
([(3,4)],5)
 => ([(3,4)],5)
 => [1]
 => [1,0,1,0]
 => 1
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 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)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([],6)
 => ([],6)
 => []
 => []
 => ? = 1
([(4,5)],6)
 => ([(4,5)],6)
 => [1]
 => [1,0,1,0]
 => 1
([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(2,5),(3,5),(4,5)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
([(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(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)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(0,5),(1,4),(1,5),(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)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 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)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 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)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1

Description

Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules.

Matching statistic: St001196
(load all 2 compositions to match this statistic)

Mapping

Mp00157: Graphs —connected complement⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001196: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 46%●distinct values known / distinct values provided: 33%

Values

([],1)
 => ([],1)
 => []
 => []
 => ? = 1
([],2)
 => ([],2)
 => []
 => []
 => ? = 1
([(0,1)],2)
 => ([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => 1
([],3)
 => ([],3)
 => []
 => []
 => ? = 1
([(1,2)],3)
 => ([(1,2)],3)
 => [1]
 => [1,0,1,0]
 => 1
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([],4)
 => ([],4)
 => []
 => []
 => ? = 1
([(2,3)],4)
 => ([(2,3)],4)
 => [1]
 => [1,0,1,0]
 => 1
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([],5)
 => ([],5)
 => []
 => []
 => ? = 1
([(3,4)],5)
 => ([(3,4)],5)
 => [1]
 => [1,0,1,0]
 => 1
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 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)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([],6)
 => ([],6)
 => []
 => []
 => ? = 1
([(4,5)],6)
 => ([(4,5)],6)
 => [1]
 => [1,0,1,0]
 => 1
([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1
([(2,5),(3,5),(4,5)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
([(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(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)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(0,5),(1,4),(1,5),(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)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 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)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 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)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1

Description

The global dimension of

$A$ minus the global dimension of

$eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ .

Matching statistic: St000790
(load all 2 compositions to match this statistic)

Mapping

Mp00157: Graphs —connected complement⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000790: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 46%●distinct values known / distinct values provided: 33%

Values

([],1)
 => ([],1)
 => []
 => []
 => ? = 1 - 1
([],2)
 => ([],2)
 => []
 => []
 => ? = 1 - 1
([(0,1)],2)
 => ([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([],3)
 => ([],3)
 => []
 => []
 => ? = 1 - 1
([(1,2)],3)
 => ([(1,2)],3)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
([],4)
 => ([],4)
 => []
 => []
 => ? = 1 - 1
([(2,3)],4)
 => ([(2,3)],4)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [1,1]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 0 = 1 - 1
([],5)
 => ([],5)
 => []
 => []
 => ? = 1 - 1
([(3,4)],5)
 => ([(3,4)],5)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [1,1]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 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]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 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)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([],6)
 => ([],6)
 => []
 => []
 => ? = 1 - 1
([(4,5)],6)
 => ([(4,5)],6)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => [1,1]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(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)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,4),(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)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(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)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 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)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 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)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1

Description

The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. Apparently, the total number of these is given in [1]. The statistic counting all pairs of distinct tunnels is the area of a Dyck path [[St000012]].

The following 224 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001471The magnitude of a Dyck path. St001578The minimal number of edges to add or remove to make a graph a line graph. St000287The number of connected components of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000553The number of blocks of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001352The number of internal nodes in the modular decomposition of a graph. St001463The number of distinct columns in the nullspace of a graph. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001708The number of pairs of vertices of different degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000455The second largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St000069The number of maximal elements of a poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001191Number of simple modules

$S$ with

$Ext_A^i(S,A)=0$ for all

$i=0,1,...,g-1$ in the corresponding Nakayama algebra

$A$ with global dimension

$g$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001722The number of minimal chains with small intervals between a binary word and the top element. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000456The monochromatic index of a connected graph. St000781The number of proper colouring schemes of a Ferrers diagram. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000326The position of the first one in a binary word after appending a 1 at the end. St000517The Kreweras number of an integer partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000629The defect of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001175The size of a partition minus the hook length of the base cell. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001371The length of the longest Yamanouchi prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001060The distinguishing index of a graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000783The side length of the largest staircase partition fitting into a partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St000143The largest repeated part of a partition. St000185The weighted size of a partition. St000225Difference between largest and smallest parts in a partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001651The Frankl number of a lattice. St001568The smallest positive integer that does not appear twice in the partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St000286The number of connected components of the complement of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000488The number of cycles of a permutation of length at most 2. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000787The number of flips required to make a perfect matching noncrossing. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000068The number of minimal elements in a poset. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001363The Euler characteristic of a graph according to Knill. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001479The number of bridges of a graph. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001730The number of times the path corresponding to a binary word crosses the base line. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001877Number of indecomposable injective modules with projective dimension 2. St001301The first Betti number of the order complex associated with the poset. 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. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001793The difference between the clique number and the chromatic number of a graph.