- FindStat

Your data matches 253 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000403
(load all 7 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
St000403: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 0
([],2)
 => ([],1)
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => 0
([],3)
 => ([],1)
 => 0
([(1,2)],3)
 => ([(1,2)],3)
 => 0
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([],4)
 => ([],1)
 => 0
([(2,3)],4)
 => ([(1,2)],3)
 => 0
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 0
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([],5)
 => ([],1)
 => 0
([(3,4)],5)
 => ([(1,2)],3)
 => 0
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 0
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 0
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => 0
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 5
([(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)
 => 10
([(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
([(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)
 => 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)
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(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)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0

Description

The Szeged index minus the Wiener index of a graph. It is known that the Szeged index is at least as much as the Wiener index. For $2$-connected graphs on $n$ vertices, the difference is at least $2n-6$. For the two individual statistics see [[St000263]] and [[St000265]].

Matching statistic: St001578
(load all 12 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001578: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 63%●distinct values known / distinct values provided: 4%

Values

([],1)
 => ([],1)
 => ([],0)
 => ? = 0
([],2)
 => ([],1)
 => ([],0)
 => ? = 0
([(0,1)],2)
 => ([(0,1)],2)
 => ([],1)
 => 0
([],3)
 => ([],1)
 => ([],0)
 => ? = 0
([(1,2)],3)
 => ([(1,2)],3)
 => ([],1)
 => 0
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([],1)
 => 0
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([],4)
 => ([],1)
 => ([],0)
 => ? = 0
([(2,3)],4)
 => ([(1,2)],3)
 => ([],1)
 => 0
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ([],1)
 => 0
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 0
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([],2)
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0
([],5)
 => ([],1)
 => ([],0)
 => ? ∊ {4,4,5,5,10}
([(3,4)],5)
 => ([(1,2)],3)
 => ([],1)
 => 0
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([],1)
 => 0
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([],1)
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 0
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([],2)
 => 0
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([],2)
 => 0
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ([],1)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 0
([(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)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {4,4,5,5,10}
([(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)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {4,4,5,5,10}
([(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)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {4,4,5,5,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
 => ? ∊ {4,4,5,5,10}
([],6)
 => ([],1)
 => ([],0)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(4,5)],6)
 => ([(1,2)],3)
 => ([],1)
 => 0
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([],1)
 => 0
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([],1)
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([],1)
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ([],1)
 => 0
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => ([],2)
 => 0
([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
([(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)
 => ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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)
 => ([(0,3),(0,5),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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)
 => ([(0,4),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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)
 => ([(0,2),(0,5),(0,7),(1,2),(1,4),(1,6),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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)
 => ([(0,4),(0,5),(1,2),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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)
 => ([(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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)
 => ([(0,3),(0,4),(0,5),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,7),(4,6),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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)
 => ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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)
 => ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,2),(1,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,6),(1,2),(1,3),(1,5),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,7),(2,4),(2,6),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,7),(1,2),(1,3),(1,6),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,3),(0,7),(1,2),(1,6),(2,4),(2,6),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(0,8),(1,2),(1,4),(1,7),(2,3),(2,6),(2,7),(3,6),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,7),(0,8),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,4),(2,3),(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)
 => ([(0,3),(0,4),(0,7),(1,2),(1,5),(1,7),(1,8),(2,3),(2,6),(2,8),(3,6),(3,8),(4,5),(4,6),(4,7),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(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)
 => ([(0,1),(0,5),(0,7),(0,8),(1,4),(1,7),(1,8),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,8),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(0,7),(0,9),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,7),(2,8),(3,6),(3,7),(3,9),(4,5),(4,6),(4,8),(4,9),(5,6),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,7),(2,3),(2,6),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,7),(1,8),(2,3),(2,6),(2,8),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,4),(2,6),(3,5),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(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)
 => ([(0,4),(0,5),(0,8),(1,2),(1,3),(1,4),(1,7),(2,3),(2,4),(2,6),(3,6),(3,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}

Description

The minimal number of edges to add or remove to make a graph a line graph.

Matching statistic: St000661

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 62%●distinct values known / distinct values provided: 4%

Values

([],1)
 => []
 => []
 => []
 => ? = 0
([],2)
 => []
 => []
 => []
 => ? ∊ {0,0}
([(0,1)],2)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,0}
([],3)
 => []
 => []
 => []
 => ? ∊ {0,0}
([(1,2)],3)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,0}
([(0,2),(1,2)],3)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([],4)
 => []
 => []
 => []
 => ? ∊ {0,0}
([(2,3)],4)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,0}
([(1,3),(2,3)],4)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([],5)
 => []
 => []
 => []
 => ? ∊ {0,4,4,5,5,10}
([(3,4)],5)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,4,4,5,5,10}
([(2,4),(3,4)],5)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([],6)
 => []
 => []
 => []
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(4,5)],6)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(3,5),(4,5)],6)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9,1]
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [11]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}

Description

The number of rises of length 3 of a Dyck path.

Matching statistic: St000980
(load all 15 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 62%●distinct values known / distinct values provided: 4%

Values

([],1)
 => []
 => []
 => []
 => ? = 0
([],2)
 => []
 => []
 => []
 => ? ∊ {0,0}
([(0,1)],2)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,0}
([],3)
 => []
 => []
 => []
 => ? ∊ {0,0}
([(1,2)],3)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,0}
([(0,2),(1,2)],3)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([],4)
 => []
 => []
 => []
 => ? ∊ {0,0}
([(2,3)],4)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,0}
([(1,3),(2,3)],4)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([],5)
 => []
 => []
 => []
 => ? ∊ {0,4,4,5,5,10}
([(3,4)],5)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,4,4,5,5,10}
([(2,4),(3,4)],5)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([],6)
 => []
 => []
 => []
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(4,5)],6)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(3,5),(4,5)],6)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9,1]
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [11]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}

Description

The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. For example, the path $111011010000$ has three peaks in positions $03, 15, 26$. The boxes below $03$ are $01,02,\textbf{12}$, the boxes below $15$ are $\textbf{12},13,14,\textbf{23},\textbf{24},\textbf{34}$, and the boxes below $26$ are $\textbf{23},\textbf{24},25,\textbf{34},35,45$. We thus obtain the four boxes in positions $12,23,24,34$ that are below at least two peaks.

Matching statistic: St001139
(load all 5 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 62%●distinct values known / distinct values provided: 4%

Values

([],1)
 => []
 => []
 => []
 => ? = 0
([],2)
 => []
 => []
 => []
 => ? ∊ {0,0}
([(0,1)],2)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,0}
([],3)
 => []
 => []
 => []
 => ? ∊ {0,0}
([(1,2)],3)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,0}
([(0,2),(1,2)],3)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([],4)
 => []
 => []
 => []
 => ? ∊ {0,0}
([(2,3)],4)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,0}
([(1,3),(2,3)],4)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([],5)
 => []
 => []
 => []
 => ? ∊ {0,4,4,5,5,10}
([(3,4)],5)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,4,4,5,5,10}
([(2,4),(3,4)],5)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([],6)
 => []
 => []
 => []
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(4,5)],6)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(3,5),(4,5)],6)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9,1]
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [11]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}

Description

The number of occurrences of hills of size 2 in a Dyck path. A hill of size two is a subpath beginning at height zero, consisting of two up steps followed by two down steps.

Matching statistic: St001141
(load all 5 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001141: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 62%●distinct values known / distinct values provided: 4%

Values

([],1)
 => []
 => []
 => []
 => ? = 0
([],2)
 => []
 => []
 => []
 => ? ∊ {0,0}
([(0,1)],2)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,0}
([],3)
 => []
 => []
 => []
 => ? ∊ {0,0}
([(1,2)],3)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,0}
([(0,2),(1,2)],3)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([],4)
 => []
 => []
 => []
 => ? ∊ {0,0}
([(2,3)],4)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,0}
([(1,3),(2,3)],4)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([],5)
 => []
 => []
 => []
 => ? ∊ {0,4,4,5,5,10}
([(3,4)],5)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {0,4,4,5,5,10}
([(2,4),(3,4)],5)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,4,4,5,5,10}
([],6)
 => []
 => []
 => []
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(4,5)],6)
 => [1]
 => [1]
 => [1,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(3,5),(4,5)],6)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9,1]
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [11]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}

Description

The number of occurrences of hills of size 3 in a Dyck path. A hill of size three is a subpath beginning at height zero, consisting of three up steps followed by three down steps.

Matching statistic: St000687
(load all 29 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000687: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 59%●distinct values known / distinct values provided: 4%

Values

([],1)
 => []
 => []
 => ? = 0
([],2)
 => []
 => []
 => ? = 0
([(0,1)],2)
 => [1]
 => [1,0]
 => 0
([],3)
 => []
 => []
 => ? = 0
([(1,2)],3)
 => [1]
 => [1,0]
 => 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1,1,0,0]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 0
([],4)
 => []
 => []
 => ? = 0
([(2,3)],4)
 => [1]
 => [1,0]
 => 0
([(1,3),(2,3)],4)
 => [1,1]
 => [1,1,0,0]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([],5)
 => []
 => []
 => ? ∊ {4,4,5,5,10}
([(3,4)],5)
 => [1]
 => [1,0]
 => 0
([(2,4),(3,4)],5)
 => [1,1]
 => [1,1,0,0]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,10}
([],6)
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(4,5)],6)
 => [1]
 => [1,0]
 => 0
([(3,5),(4,5)],6)
 => [1,1]
 => [1,1,0,0]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [11]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}

Description

The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. In this expression, $I$ is the direct sum of all injective non-projective indecomposable modules and $P$ is the direct sum of all projective non-injective indecomposable modules. This statistic was discussed in [Theorem 5.7, 1].

Matching statistic: St001125
(load all 2 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001125: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 59%●distinct values known / distinct values provided: 4%

Values

([],1)
 => []
 => []
 => ? = 0
([],2)
 => []
 => []
 => ? = 0
([(0,1)],2)
 => [1]
 => [1,0]
 => 0
([],3)
 => []
 => []
 => ? = 0
([(1,2)],3)
 => [1]
 => [1,0]
 => 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1,1,0,0]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 0
([],4)
 => []
 => []
 => ? = 0
([(2,3)],4)
 => [1]
 => [1,0]
 => 0
([(1,3),(2,3)],4)
 => [1,1]
 => [1,1,0,0]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([],5)
 => []
 => []
 => ? ∊ {4,4,5,5,10}
([(3,4)],5)
 => [1]
 => [1,0]
 => 0
([(2,4),(3,4)],5)
 => [1,1]
 => [1,1,0,0]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {4,4,5,5,10}
([],6)
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(4,5)],6)
 => [1]
 => [1,0]
 => 0
([(3,5),(4,5)],6)
 => [1,1]
 => [1,1,0,0]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [11]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}

Description

The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra.

Matching statistic: St000966
(load all 18 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000966: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 59%●distinct values known / distinct values provided: 4%

Values

([],1)
 => []
 => []
 => []
 => ? = 0
([],2)
 => []
 => []
 => []
 => ? = 0
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([],3)
 => []
 => []
 => []
 => ? = 0
([(1,2)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([],4)
 => []
 => []
 => []
 => ? = 0
([(2,3)],4)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(1,3),(2,3)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([],5)
 => []
 => []
 => []
 => ? ∊ {4,4,5,5,10}
([(3,4)],5)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(2,4),(3,4)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {4,4,5,5,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {4,4,5,5,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {4,4,5,5,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {4,4,5,5,10}
([],6)
 => []
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(4,5)],6)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(3,5),(4,5)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [11]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}

Description

Number of peaks minus the global dimension of the corresponding LNakayama algebra.

Matching statistic: St001025
(load all 26 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001025: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 59%●distinct values known / distinct values provided: 4%

Values

([],1)
 => []
 => []
 => []
 => ? = 0
([],2)
 => []
 => []
 => []
 => ? = 0
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([],3)
 => []
 => []
 => []
 => ? = 0
([(1,2)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([],4)
 => []
 => []
 => []
 => ? = 0
([(2,3)],4)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(1,3),(2,3)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([],5)
 => []
 => []
 => []
 => ? ∊ {4,4,5,5,10}
([(3,4)],5)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(2,4),(3,4)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {4,4,5,5,10}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {4,4,5,5,10}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {4,4,5,5,10}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {4,4,5,5,10}
([],6)
 => []
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(4,5)],6)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(3,5),(4,5)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [11]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,8,8,8,8,8,8,8,9,9,10,10,10,10,10,10,11,11,12,12,12,12,13,14,15,15,15,17,18,18,18,19,20,21,21,23,24,27,29,30,32,34}

Description

Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path.

The following 243 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St000379The number of Hamiltonian cycles in a graph. St000768The number of peaks in an integer composition. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St000449The number of pairs of vertices of a graph with distance 4. St000790The number of pairs of centered tunnels, one strictly containing the other, of a 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. St001584The area statistic between a Dyck path and its bounce path. St000376The bounce deficit of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000422The energy of a graph, if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000369The dinv deficit of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001307The number of induced stars on four vertices in a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001330The hat guessing number of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. 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. 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. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St000447The number of pairs of vertices of a graph with distance 3. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000312The number of leaves in a graph. St000313The number of degree 2 vertices of a graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two 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. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001649The length of a longest trail in a graph. St000617The number of global maxima of a Dyck path. St001386The number of prime labellings of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000916The packing number of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. 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. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000448The number of pairs of vertices of a graph with distance 2. St000552The number of cut vertices of a graph. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001323The independence gap of a graph. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001374The Padmakar-Ivan 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. St001799The number of proper separations of a graph. St000095The number of triangles of a graph. St000274The number of perfect matchings of a graph. St000311The number of vertices of odd degree in a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000350The sum of the vertex degrees of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000637The length of the longest cycle in a graph. St000915The Ore degree of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001341The number of edges in the center of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001395The number of strictly unfriendly partitions of a graph. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001479The number of bridges of a graph. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001783The number of odd automorphisms of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001826The maximal number of leaves on a vertex of a graph. St001869The maximum cut size of a graph. St001871The number of triconnected components of a graph. St001969The difference in the number of possibilities of choosing a pair of negative eigenvalues and the signature of a graph. St001973The Gromov width of a graph. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. 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. St000069The number of maximal elements of a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001890The maximum magnitude of the Möbius function of a poset. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001846The number of elements which do not have a complement in the lattice. 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. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000455The second largest eigenvalue of a graph if it is integral. 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. 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. St001214The aft of an integer partition. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001485The modular major index of a binary word. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000143The largest repeated part of a partition. St000185The weighted size of a partition. St000225Difference between largest and smallest parts in a 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. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St001845The number of join irreducibles minus the rank of a lattice. 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. 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. 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 $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. 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. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000068The number of minimal elements in a poset. St001271The competition number of a graph. 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. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001730The number of times the path corresponding to a binary word crosses the base line. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001301The first Betti number of the order complex associated with the poset. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000944The 3-degree of an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001793The difference between the clique number and the chromatic number of a graph. St000553The number of blocks of a graph. 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. St001740The number of graphs with the same symmetric edge polytope as the given graph.