- FindStat

Your data matches 169 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001797
(load all 10 compositions to match this statistic)

Mapping

St001797: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of overfull subgraphs of a graph. A subgraph

$H=(V, E)$ of a graph

$G$ is overfull if

$|E| > \Delta(G) \lfloor\frac{V}{2}\rfloor$ . This statistic counts the number of subsets

$E$ of the edge set of

$G$ , such that the edge-induced subgraph is overfull.

Matching statistic: St001251

Mapping

Mp00264: Graphs —delete endpoints⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts of a partition that are not congruent 1 modulo 3.

Matching statistic: St001845
(load all 5 compositions to match this statistic)

Mapping

Mp00266: Graphs —connected vertex partitions⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => ([],1)
 => 0
([],2)
 => ([],1)
 => ([],1)
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([],3)
 => ([],1)
 => ([],1)
 => 0
([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
([],4)
 => ([],1)
 => ([],1)
 => 0
([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ([],1)
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ([],1)
 => 0
([],5)
 => ([],1)
 => ([],1)
 => 0
([(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ([],1)
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ([],1)
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ([],1)
 => 0
([],6)
 => ([],1)
 => ([],1)
 => 0
([(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
([(1,2),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ([],1)
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ([],1)
 => 0
([(0,5),(1,4),(2,3)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ([],1)
 => 0
([],7)
 => ([],1)
 => ([],1)
 => 0
([(5,6)],7)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
([(2,3),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ([],1)
 => 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ([],1)
 => 0

Description

The number of join irreducibles minus the rank of a lattice. A lattice is join-extremal, if this statistic is

$0$ .

Matching statistic: St001141
(load all 7 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001141: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

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

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: St000752
(load all 2 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000752: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%

Values

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

Description

The Grundy value for the game 'Couples are forever' on an integer partition. Two players alternately choose a part of the partition greater than two, and split it into two parts. The player facing a partition with all parts at most two looses.

Matching statistic: St001137
(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
St001137: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%

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

Description

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

Matching statistic: St000661
(load all 5 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000688

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St000688: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%

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

Description

The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. The global dimension is given by [[St000684]] and the dominant dimension is given by [[St000685]]. To every Dyck path there is an LNakayama algebra associated as described in [[St000684]]. Dyck paths for which the global dimension and the dominant dimension of the the LNakayama algebra coincide and both dimensions at least

$2$ correspond to the LNakayama algebras that are higher Auslander algebras in the sense of [1].

Matching statistic: St001021

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St001021: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%

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

Description

Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001089

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St001089: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%

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

Description

Number 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.

The following 159 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. 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. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. 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. St001651The Frankl number of a lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000456The monochromatic index of a connected graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. 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. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer 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. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001250The number of parts of a partition that are not congruent 0 modulo 3. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001933The largest multiplicity of a part in an integer partition. St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St000544The cop number of a graph. St001890The maximum magnitude of the Möbius function of a poset. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001363The Euler characteristic of a graph according to Knill. St001875The number of simple modules with projective dimension at most 1. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001128The exponens consonantiae of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000997The even-odd crank of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001568The smallest positive integer that does not appear twice in the partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001118The acyclic chromatic index of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001095The number of non-isomorphic posets with precisely one further covering relation. St000914The sum of the values of the Möbius function of a poset. St001330The hat guessing number of a graph.