- FindStat

Your data matches 210 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000527
(load all 2 compositions to match this statistic)

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.

Matching statistic: St000010

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000097

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The order of the largest clique of the graph. A clique in a graph

$G$ is a subset

$U \subseteq V(G)$ such that any pair of vertices in

$U$ are adjacent. I.e. the subgraph induced by

$U$ is a complete graph.

Matching statistic: St000098
(load all 2 compositions to match this statistic)

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

Matching statistic: St000533
(load all 2 compositions to match this statistic)

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000533: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The minimum of the number of parts and the size of the first part of an integer partition. This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.

Matching statistic: St000093

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number

$\alpha(G)$ of

$G$ .

Matching statistic: St000147

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000225

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Difference between largest and smallest parts in a partition.

Matching statistic: St000319

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The spin of an integer partition. The Ferrers shape of an integer partition

$\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of

$\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let

$\lambda = (5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$ The first strip

$(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses

$4$ times, the second strip

$(4,3,3,1) \setminus (2,2)$ crosses

$3$ times, the strip

$(2,2) \setminus (1)$ crosses

$1$ time, and the remaining strip

$(1) \setminus ()$ does not cross. This yields the spin of

$(5,5,4,4,2,1)$ to be

$4+3+1 = 8$ .

Matching statistic: St000320

Mapping

Mp00209: Permutations —pattern poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dinv adjustment of an integer partition. The Ferrers shape of an integer partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For

$0 \leq j < \lambda_1$ let

$n_j$ be the length of the border strip starting at

$(\lambda_1-j,0)$ . The dinv adjustment is then defined by

$\sum_{j:n_j > 0}(\lambda_1-1-j).$ The following example is taken from Appendix B in [2]: Let

$\lambda=(5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$ and we obtain

$(n_0,\ldots,n_4) = (10,7,0,3,1)$ . The dinv adjustment is thus

$4+3+1+0 = 8$ .

The following 200 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001252Half the sum of the even parts of a partition. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001388The number of non-attacking neighbors of a permutation. St000008The major index of the composition. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000785The number of distinct colouring schemes of a graph. St001029The size of the core of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001883The mutual visibility number of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001644The dimension of a graph. St001792The arboricity of a graph. St001962The proper pathwidth of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001323The independence gap of a graph. St001638The book thickness of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000937The number of positive values of the symmetric group character corresponding to the partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000077The number of boxed and circled entries. St001668The number of points of the poset minus the width of the poset. St000307The number of rowmotion orbits of a poset. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001060The distinguishing index of a graph. St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . St001200The number of simple modules in

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

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St000259The diameter of a connected graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000450The number of edges minus the number of vertices plus 2 of a graph. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001510The number of self-evacuating linear extensions of a finite poset. St001734The lettericity of a graph. St000632The jump number of the poset. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001642The Prague dimension of a graph. St000274The number of perfect matchings of a graph. St000310The minimal degree of a vertex of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001820The size of the image of the pop stack sorting operator. St001118The acyclic chromatic index of a graph. St000260The radius of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001645The pebbling number of a connected graph. St001651The Frankl number of a lattice. St000438The position of the last up step in a Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St001669The number of single rises in a Dyck path. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000762The sum of the positions of the weak records of an integer composition. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000455The second largest eigenvalue of a graph if it is integral. St001624The breadth of a lattice. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St001282The number of graphs with the same chromatic polynomial. St001333The cardinality of a minimal edge-isolating set of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000552The number of cut vertices of a graph. St000636The hull number of a graph. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001691The number of kings in a graph. St001793The difference between the clique number and the chromatic number of a graph. St000264The girth of a graph, which is not a tree. St001877Number of indecomposable injective modules with projective dimension 2. St001875The number of simple modules with projective dimension at most 1. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000899The maximal number of repetitions of an integer composition. St000902 The minimal number of repetitions of an integer composition. St000954Number of times the corresponding LNakayama algebra has

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

$i>0$ . St001188The number of simple modules

$S$ with grade

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

$A$ corresponding to the Dyck path. St001196The global dimension of

$A$ minus the global dimension of

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

$eA$ . St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001395The number of strictly unfriendly partitions of a graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000367The number of simsun double descents of a permutation. St000422The energy of a graph, if it is integral. St000768The number of peaks in an integer composition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001470The cyclic holeyness of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001857The number of edges in the reduced word graph of a signed permutation. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000649The number of 3-excedences of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001926Sparre Andersen's position of the maximum of a signed permutation. St000741The Colin de Verdière graph invariant. St001487The number of inner corners of a skew partition. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001488The number of corners of a skew partition. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000477The weight of a partition according to Alladi. St000706The product of the factorials of the multiplicities of an integer partition. St000997The even-odd crank of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. 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. St000102The charge of a semistandard tableau. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000736The last entry in the first row of a semistandard tableau. St000739The first entry in the last row of a semistandard tableau. St001208The number of connected components of the quiver of

$A/T$ when

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

$A$ of

$K[x]/(x^n)$ . St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001569The maximal modular displacement of a permutation. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001828The Euler characteristic of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St000095The number of triangles of a graph. St000101The cocharge of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000315The number of isolated vertices of a graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001625The Möbius invariant of a lattice. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001783The number of odd automorphisms of a graph. St001845The number of join irreducibles minus the rank of a lattice. St001856The number of edges in the reduced word graph of a permutation. St001871The number of triconnected components of a graph. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one.