- FindStat

Your data matches 299 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000310
(load all 6 compositions to match this statistic)

Mapping

St000310: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The minimal degree of a vertex of a graph.

Matching statistic: St000261
(load all 4 compositions to match this statistic)

Mapping

Mp00203: Graphs —cone⟶ Graphs
St000261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The edge connectivity of a graph. This is the minimum number of edges that has to be removed to make the graph disconnected.

Matching statistic: St001216

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001216: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module.

Matching statistic: St001274

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001274: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indecomposable injective modules with projective dimension equal to two.

Matching statistic: St001089

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001089: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0 = 1 - 1
([],2)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
([],3)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
([(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
([(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 = 2 - 1
([],4)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3 = 4 - 1
([(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 3 - 1
([],5)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4 = 5 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 2 = 3 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? ∊ {1,1,2,2} - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => ? ∊ {1,1,2,2} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? ∊ {1,1,2,2} - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? ∊ {1,1,2,2} - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 3 = 4 - 1

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.

Matching statistic: St000454
(load all 27 compositions to match this statistic)

Mapping

Mp00264: Graphs —delete endpoints⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],2)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,1)],2)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(1,2)],3)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 0 = 1 - 1
([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(2,3)],4)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(1,3),(2,3)],4)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 0 = 1 - 1
([],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
([(3,4)],5)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(2,4),(3,4)],5)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,3} - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,3} - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,3} - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,3} - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,3} - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,3} - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(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)
 => ([],4)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,3} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,3} - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => 0 = 1 - 1

Description

The largest eigenvalue of a graph if it is integral. If a graph is

$d$ -regular, then its largest eigenvalue equals

$d$ . One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

Matching statistic: St000442

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 80%

Values

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

Description

The maximal area to the right of an up step of a Dyck path.

Matching statistic: St000524

Mapping

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
Mp00243: Graphs —weak duplicate order⟶ Posets
St000524: Posets ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 80%

Values

([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 1
([],2)
 => ([],2)
 => ([(0,1)],2)
 => ([],2)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? = 2
([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 1
([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([],2)
 => 1
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ([],1)
 => ? = 3
([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 1
([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],3)
 => 1
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],2)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2)],4)
 => 3
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([],2)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {2,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {2,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => ([],1)
 => ? ∊ {2,2,4}
([],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => 1
([(3,4)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],4)
 => 1
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],3)
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => 3
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(2,3)],4)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3)],5)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],2)
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],2)
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],5)
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {1,1,2,2,2,3,5}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {1,1,2,2,2,3,5}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],2)
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {1,1,2,2,2,3,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {1,1,2,2,2,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {1,1,2,2,2,3,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {1,1,2,2,2,3,5}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],1)
 => ? ∊ {1,1,2,2,2,3,5}

Description

The number of posets with the same order polynomial. The order polynomial of a poset

$P$ is the polynomial

$S$ such that

$S(m)$ is the number of order-preserving maps from

$P$ to

$\{1,\dots,m\}$ . See sections 3.12 and 3.15 of [1].

Matching statistic: St000771
(load all 47 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 80%

Values

([],1)
 => [1] => [1] => ([],1)
 => 1
([],2)
 => [2] => [2] => ([],2)
 => ? = 2
([(0,1)],2)
 => [1,1] => [1,1] => ([(0,1)],2)
 => 1
([],3)
 => [3] => [3] => ([],3)
 => ? = 3
([(1,2)],3)
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
([(0,2),(1,2)],3)
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([],4)
 => [4] => [4] => ([],4)
 => ? ∊ {1,2,3,4}
([(2,3)],4)
 => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
([(1,3),(2,3)],4)
 => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,2,3,4}
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,2,3,4}
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,2,3,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([],5)
 => [5] => [5] => ([],5)
 => ? ∊ {1,1,2,3,3,3,3,4,5}
([(3,4)],5)
 => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(2,4),(3,4)],5)
 => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(1,4),(2,3)],5)
 => [3,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,3,3,3,3,4,5}
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,3,3,3,3,4,5}
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,3,3,3,3,4,5}
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,3,3,3,3,4,5}
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,3,3,3,3,4,5}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,3,3,3,3,4,5}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,3,3,3,3,4,5}
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4

Description

The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums

$0$ , which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian

$\left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right).$ Its eigenvalues are

$0,4,4,6$ , so the statistic is

$2$ . The path on four vertices has eigenvalues

$0, 4.7\dots, 6, 9.2\dots$ and therefore statistic

$1$ .

Matching statistic: St001199
(load all 11 compositions to match this statistic)

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 80%

Values

([],1)
 => [1] => [1] => [1,0]
 => ? = 1
([],2)
 => [2] => [1,1] => [1,0,1,0]
 => 1
([(0,1)],2)
 => [1,1] => [2] => [1,1,0,0]
 => ? = 2
([],3)
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 2
([(1,2)],3)
 => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 1
([(0,2),(1,2)],3)
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => ? = 3
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
([],4)
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
([(2,3)],4)
 => [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
([(1,3),(2,3)],4)
 => [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
([(0,3),(1,2)],4)
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {3,4}
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {3,4}
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
([],5)
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 4
([(3,4)],5)
 => [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
([(2,4),(3,4)],5)
 => [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
([(1,4),(2,3)],5)
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,1,2,2,3,3,3,4,5}
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,1,2,2,3,3,3,4,5}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,1,2,2,3,3,3,4,5}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,1,2,2,3,3,3,4,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,1,2,2,3,3,3,4,5}
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,1,2,2,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,1,2,2,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,1,2,2,3,3,3,4,5}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,1,2,2,3,3,3,4,5}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,1,2,2,3,3,3,4,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 3

Description

The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

The following 289 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000776The maximal multiplicity of an eigenvalue in a graph. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000741The Colin de Verdière graph invariant. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St001118The acyclic chromatic index of a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001570The minimal number of edges to add to make a graph Hamiltonian. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001330The hat guessing number of a graph. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001933The largest multiplicity of a part in an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St001389The number of partitions of the same length below the given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001060The distinguishing index of a graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000003The number of standard Young tableaux of the partition. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000010The length of the partition. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000079The number of alternating sign matrices for a given Dyck path. St000108The number of partitions contained in the given partition. St000120The number of left tunnels of a Dyck path. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000160The multiplicity of the smallest part of a partition. St000182The number of permutations whose cycle type is the given integer partition. St000228The size of a partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000288The number of ones in a binary word. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000294The number of distinct factors of a binary word. St000297The number of leading ones in a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000439The position of the first down step of a Dyck path. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000459The hook length of the base cell of a partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000517The Kreweras number of an integer partition. St000518The number of distinct subsequences in a binary word. St000519The largest length of a factor maximising the subword complexity. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000543The size of the conjugacy class of a binary word. St000548The number of different non-empty partial sums of an integer partition. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000655The length of the minimal rise of a Dyck path. St000674The number of hills of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000759The smallest missing part in an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000867The sum of the hook lengths in the first row of an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000935The number of ordered refinements of an integer partition. St000947The major index east count of a Dyck path. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St000982The length of the longest constant subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001191Number of simple modules

$S$ with

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

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

$A$ with global dimension

$g$ . 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

$eA$ . St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call 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. St001203We associate to 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 Dyck path as follows: St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001432The order dimension of the partition. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001437The flex of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001471The magnitude of a Dyck path. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001481The minimal height of a peak of a Dyck path. St001488The number of corners of a skew partition. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001614The cyclic permutation representation number of a skew partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001721The degree of a binary word. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001780The order of promotion on the set of standard tableaux of given shape. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001884The number of borders of a binary word. St001885The number of binary words with the same proper border set. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. 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. St001955The number of natural descents for set-valued two row standard Young tableaux. St001959The product of the heights of the peaks of a Dyck path. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001621The number of atoms of a lattice. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001378The product of the cohook lengths of the integer partition. St001563The value of the power-sum symmetric function evaluated at 1. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St001545The second Elser number of a connected graph. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000815The number of semistandard Young tableaux of partition weight of given shape. 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. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000264The girth of a graph, which is not a tree. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles 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. St000640The rank of the largest boolean interval in a poset. St000681The Grundy value of Chomp on Ferrers diagrams. St000770The major index of an integer partition when read from bottom to top. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000910The number of maximal chains of minimal length in a poset. St001890The maximum magnitude of the Möbius function of a poset. St000477The weight of a partition according to Alladi. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000707The product of the factorials of the parts. St000420The number of Dyck paths that are weakly above a Dyck path. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000675The number of centered multitunnels of a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000422The energy of a graph, if it is integral. St001875The number of simple modules with projective dimension at most 1. St000455The second largest eigenvalue of a graph if it is integral. St001644The dimension of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000618The number of self-evacuating tableaux of given shape. 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. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for 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

$eA$ . 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

$eA$ . St001645The pebbling number of a connected graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001651The Frankl number of a lattice. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. 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.