Your data matches 36 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000327
(load all 2 compositions to match this statistic)
Mapping
St000327: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
 => 0
([(0,1)],2)
 => 1
([],3)
 => 0
([(1,2)],3)
 => 1
([(0,1),(0,2)],3)
 => 2
([(0,2),(2,1)],3)
 => 2
([(0,2),(1,2)],3)
 => 2
([],4)
 => 0
([(2,3)],4)
 => 1
([(1,2),(1,3)],4)
 => 2
([(0,1),(0,2),(0,3)],4)
 => 3
([(0,2),(0,3),(3,1)],4)
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
([(1,2),(2,3)],4)
 => 2
([(0,3),(3,1),(3,2)],4)
 => 3
([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(3,2)],4)
 => 3
([(0,3),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(1,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
([(0,3),(2,1),(3,2)],4)
 => 3
([(0,3),(1,2),(2,3)],4)
 => 3
([],5)
 => 0
([(3,4)],5)
 => 1
([(2,3),(2,4)],5)
 => 2
([(1,2),(1,3),(1,4)],5)
 => 3
([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
([(0,2),(0,3),(0,4),(4,1)],5)
 => 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 6
([(1,3),(1,4),(4,2)],5)
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => 4
([(1,2),(1,3),(2,4),(3,4)],5)
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 6
([(2,3),(3,4)],5)
 => 2
([(1,4),(4,2),(4,3)],5)
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => 4
([(2,4),(3,4)],5)
 => 2
([(1,4),(2,4),(4,3)],5)
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => 4
([(1,4),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
([(0,4),(1,4),(2,3)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 5
Description
The number of cover relations in a poset. Equivalently, this is also the number of edges in the Hasse diagram [1].
Matching statistic: St000081
(load all 2 compositions to match this statistic)
Mapping
Mp00074: Posets —to graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
 => ([],2)
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => 1
([],3)
 => ([],3)
 => 0
([(1,2)],3)
 => ([(1,2)],3)
 => 1
([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
([],4)
 => ([],4)
 => 0
([(2,3)],4)
 => ([(2,3)],4)
 => 1
([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
([],5)
 => ([],5)
 => 0
([(3,4)],5)
 => ([(3,4)],5)
 => 1
([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => 2
([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 6
([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 4
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 6
([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 2
([(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 2
([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 5
Description
The number of edges of a graph.
Matching statistic: St000095
Mapping
Mp00074: Posets —to graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000095: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
 => ([],2)
 => ([(0,2),(1,2)],3)
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([],3)
 => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(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)
 => 4
([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(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)
 => 4
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0
([(3,4)],5)
 => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(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)
 => 4
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(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)
 => 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(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)
 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6
([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(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)
 => 4
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(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)
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(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)
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(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)
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(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)
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(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)
 => 6
([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(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)
 => 4
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(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)
 => 4
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(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)
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(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)
 => 4
([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(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)
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(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)
 => 5
Description
The number of triangles of a graph. A triangle $T$ of a graph $G$ is a collection of three vertices $\{u,v,w\} \in G$ such that they form $K_3$, the complete graph on three vertices.
Matching statistic: St000142
Mapping
Mp00074: Posets —to graph⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
 => ([],2)
 => [1,1]
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 1
([],3)
 => ([],3)
 => [1,1,1]
 => 0
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 1
([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([],4)
 => ([],4)
 => [1,1,1,1]
 => 0
([(2,3)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => 1
([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 4
([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 4
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([],5)
 => ([],5)
 => [1,1,1,1,1]
 => 0
([(3,4)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => 1
([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,2,2]
 => 6
([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => 4
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,1]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,2,2]
 => 6
([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
Description
The number of even parts of a partition.
Matching statistic: St000228
(load all 4 compositions to match this statistic)
Mapping
Mp00074: Posets —to graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
 => ([],2)
 => []
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [1]
 => 1
([],3)
 => ([],3)
 => []
 => 0
([(1,2)],3)
 => ([(1,2)],3)
 => [1]
 => 1
([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => [2]
 => 2
([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => [2]
 => 2
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2]
 => 2
([],4)
 => ([],4)
 => []
 => 0
([(2,3)],4)
 => ([(2,3)],4)
 => [1]
 => 1
([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => [2]
 => 2
([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => 3
([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 4
([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2]
 => 2
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => 3
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2]
 => 2
([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => 3
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 4
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 3
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 3
([],5)
 => ([],5)
 => []
 => 0
([(3,4)],5)
 => ([(3,4)],5)
 => [1]
 => 1
([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => [2]
 => 2
([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => 3
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 4
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 6
([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => 4
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 6
([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2]
 => 2
([(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 4
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2]
 => 2
([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 4
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 4
([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 5
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000459
(load all 4 compositions to match this statistic)
Mapping
Mp00074: Posets —to graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
 => ([],2)
 => []
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [1]
 => 1
([],3)
 => ([],3)
 => []
 => 0
([(1,2)],3)
 => ([(1,2)],3)
 => [1]
 => 1
([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => [2]
 => 2
([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => [2]
 => 2
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2]
 => 2
([],4)
 => ([],4)
 => []
 => 0
([(2,3)],4)
 => ([(2,3)],4)
 => [1]
 => 1
([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => [2]
 => 2
([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => 3
([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 4
([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2]
 => 2
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => 3
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2]
 => 2
([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [3]
 => 3
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 4
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 3
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 3
([],5)
 => ([],5)
 => []
 => 0
([(3,4)],5)
 => ([(3,4)],5)
 => [1]
 => 1
([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => [2]
 => 2
([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => 3
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 4
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 6
([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [3]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => 4
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 6
([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2]
 => 2
([(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 4
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2]
 => 2
([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 4
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [3]
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 4
([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 5
Description
The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000566
Mapping
Mp00074: Posets —to graph⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
 => ([],2)
 => [1,1]
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 1
([],3)
 => ([],3)
 => [1,1,1]
 => 0
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 1
([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([],4)
 => ([],4)
 => [1,1,1,1]
 => 0
([(2,3)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => 1
([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 4
([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 4
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([],5)
 => ([],5)
 => [1,1,1,1,1]
 => 0
([(3,4)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => 1
([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,2,2]
 => 6
([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => 4
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,1]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,2,2]
 => 6
([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is $$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St001251
Mapping
Mp00074: Posets —to graph⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
 => ([],2)
 => [1,1]
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 1
([],3)
 => ([],3)
 => [1,1,1]
 => 0
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 1
([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([],4)
 => ([],4)
 => [1,1,1,1]
 => 0
([(2,3)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => 1
([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 4
([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 4
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([],5)
 => ([],5)
 => [1,1,1,1,1]
 => 0
([(3,4)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => 1
([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,2,2]
 => 6
([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => 4
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,1]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,2,2]
 => 6
([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001252
Mapping
Mp00074: Posets —to graph⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
 => ([],2)
 => [1,1]
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 1
([],3)
 => ([],3)
 => [1,1,1]
 => 0
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 1
([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([],4)
 => ([],4)
 => [1,1,1,1]
 => 0
([(2,3)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => 1
([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 4
([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 4
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([],5)
 => ([],5)
 => [1,1,1,1,1]
 => 0
([(3,4)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => 1
([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,2,2]
 => 6
([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => 4
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,1]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,2,2]
 => 6
([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
Description
Half the sum of the even parts of a partition.
Matching statistic: St001280
Mapping
Mp00074: Posets —to graph⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
 => ([],2)
 => [1,1]
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 1
([],3)
 => ([],3)
 => [1,1,1]
 => 0
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 1
([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([],4)
 => ([],4)
 => [1,1,1,1]
 => 0
([(2,3)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => 1
([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 4
([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 4
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([],5)
 => ([],5)
 => [1,1,1,1,1]
 => 0
([(3,4)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => 1
([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,2,2]
 => 6
([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => 4
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,1]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,2,2]
 => 6
([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 5
Description
The number of parts of an integer partition that are at least two.
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001311The cyclomatic number of a graph. St001341The number of edges in the center of a graph. St001657The number of twos in an integer partition. St000345The number of refinements of a partition. St000450The number of edges minus the number of vertices plus 2 of a graph. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000185The weighted size of a partition. St000448The number of pairs of vertices of a graph with distance 2. St001176The size of a partition minus its first part. St001646The number of edges that can be added without increasing the maximal degree of a graph. 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. 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. St001746The coalition number of a graph. St000321The number of integer partitions of n that are dominated by an integer partition. St001622The number of join-irreducible elements of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001621The number of atoms of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001330The hat guessing number of a graph.