- FindStat

Your data matches 301 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000260
(load all 27 compositions to match this statistic)

Mapping

Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The radius of a connected graph. This is the minimum eccentricity of any vertex.

Matching statistic: St000272
(load all 2 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The treewidth of a graph. A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.

Matching statistic: St000319

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ 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
{{1,2}}
 => [2,1] => [2,1] => [2]
 => 1
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [2,1]
 => 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1]
 => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [2,1]
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,2,3,1] => [2,1,1]
 => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => [2,1,1]
 => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,3,2,1] => [2,2]
 => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,2]
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,1]
 => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [2,2]
 => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [2,1,1]
 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [2,1,1]
 => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [2,1,1]
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,1,1,1]
 => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,2,3,1,5] => [2,1,1,1]
 => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,2,4,3,1] => [2,2,1]
 => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,2,1,5,4] => [2,2,1]
 => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,2,1,4,5] => [2,1,1,1]
 => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [5,3,2,4,1] => [2,2,1]
 => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,5,3,1,2] => [2,2,1]
 => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,3,2,1,5] => [2,2,1]
 => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
 => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
 => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
 => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,3,2,4,1] => [2,2,1]
 => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => [2,2,1]
 => 1
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,3,4,2] => [2,1,1,1]
 => 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
 => 1
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
 => 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => 1
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
 => 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
 => 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [6,2,3,4,5,1] => [2,1,1,1,1]
 => 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => [5,2,3,4,1,6] => [2,1,1,1,1]
 => 1
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [6,2,3,5,4,1] => [2,2,1,1]
 => 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [4,2,3,1,6,5] => [2,2,1,1]
 => 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [4,2,3,1,5,6] => [2,1,1,1,1]
 => 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

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ 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
{{1,2}}
 => [2,1] => [2,1] => [2]
 => 1
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [2,1]
 => 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1]
 => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [2,1]
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,2,3,1] => [2,1,1]
 => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => [2,1,1]
 => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,3,2,1] => [2,2]
 => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,2]
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,1]
 => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [2,2]
 => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [2,1,1]
 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [2,1,1]
 => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [2,1,1]
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,1,1,1]
 => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,2,3,1,5] => [2,1,1,1]
 => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,2,4,3,1] => [2,2,1]
 => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,2,1,5,4] => [2,2,1]
 => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,2,1,4,5] => [2,1,1,1]
 => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [5,3,2,4,1] => [2,2,1]
 => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,5,3,1,2] => [2,2,1]
 => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,3,2,1,5] => [2,2,1]
 => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
 => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
 => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
 => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,3,2,4,1] => [2,2,1]
 => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => [2,2,1]
 => 1
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,3,4,2] => [2,1,1,1]
 => 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
 => 1
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
 => 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => 1
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
 => 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
 => 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [6,2,3,4,5,1] => [2,1,1,1,1]
 => 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => [5,2,3,4,1,6] => [2,1,1,1,1]
 => 1
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [6,2,3,5,4,1] => [2,2,1,1]
 => 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [4,2,3,1,6,5] => [2,2,1,1]
 => 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [4,2,3,1,5,6] => [2,1,1,1,1]
 => 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

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

Matching statistic: St000480

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1] => [1]
 => 0
{{1,2}}
 => [2,1] => [2,1] => [2]
 => 1
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [2,1]
 => 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1]
 => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [2,1]
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,2,3,1] => [2,1,1]
 => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => [2,1,1]
 => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,3,2,1] => [2,2]
 => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,2]
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,1]
 => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [2,2]
 => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [2,1,1]
 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [2,1,1]
 => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [2,1,1]
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,1,1,1]
 => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,2,3,1,5] => [2,1,1,1]
 => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,2,4,3,1] => [2,2,1]
 => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,2,1,5,4] => [2,2,1]
 => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,2,1,4,5] => [2,1,1,1]
 => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [5,3,2,4,1] => [2,2,1]
 => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,5,3,1,2] => [2,2,1]
 => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,3,2,1,5] => [2,2,1]
 => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
 => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
 => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
 => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,3,2,4,1] => [2,2,1]
 => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => [2,2,1]
 => 1
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,3,4,2] => [2,1,1,1]
 => 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
 => 1
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
 => 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => 1
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
 => 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
 => 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [6,2,3,4,5,1] => [2,1,1,1,1]
 => 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => [5,2,3,4,1,6] => [2,1,1,1,1]
 => 1
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [6,2,3,5,4,1] => [2,2,1,1]
 => 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [4,2,3,1,6,5] => [2,2,1,1]
 => 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [4,2,3,1,5,6] => [2,1,1,1,1]
 => 1

Description

The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for

$n\neq 2$ ) can cover is

$\frac{1}{2}(\sqrt{1+8n}-3)$ and an element which covers this number of elements is given by

$(c+i,c,c-1,\dots,3,2,1)$ , where

$1\leq i\leq c+2$ .

Matching statistic: St000535
(load all 7 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000535: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The rank-width of a graph.

Matching statistic: St000651
(load all 2 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1]
 => [[1]]
 => [1] => 0
{{1,2}}
 => [2]
 => [[1,2]]
 => [1,2] => 1
{{1,2,3}}
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 1
{{1,2},{3}}
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
{{1},{2,3}}
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
{{1,2,3,4}}
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
{{1,2,3},{4}}
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
{{1,2,4},{3}}
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
{{1,2},{3,4}}
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 1
{{1,2},{3},{4}}
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
{{1,4},{2,3}}
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 1
{{1},{2,3,4}}
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
{{1},{2,3},{4}}
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
{{1},{2},{3,4}}
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
{{1,2,3,4,5}}
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 1
{{1,2,3,4},{5}}
 => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
{{1,2,3,5},{4}}
 => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
{{1,2,3},{4,5}}
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 1
{{1,2,3},{4},{5}}
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
{{1,2,4,5},{3}}
 => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
{{1,2,4},{3,5}}
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 1
{{1,2,4},{3},{5}}
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
{{1,2,5},{3,4}}
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 1
{{1,2},{3,4,5}}
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 1
{{1,2},{3,4},{5}}
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 1
{{1,2,5},{3},{4}}
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
{{1,2},{3,5},{4}}
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 1
{{1,2},{3},{4,5}}
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
{{1,3,4},{2,5}}
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 1
{{1,4,5},{2,3}}
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 1
{{1,4},{2,3,5}}
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 1
{{1,4},{2,3},{5}}
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 1
{{1,5},{2,3,4}}
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 1
{{1},{2,3,4,5}}
 => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
{{1},{2,3,4},{5}}
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
{{1,5},{2,3},{4}}
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 1
{{1},{2,3,5},{4}}
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
{{1},{2,3},{4,5}}
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
{{1,5},{2},{3,4}}
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 1
{{1},{2,5},{3,4}}
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 1
{{1},{2},{3,4,5}}
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
{{1,2,3,4,5,6}}
 => [6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => 1
{{1,2,3,4,5},{6}}
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => 1
{{1,2,3,4,6},{5}}
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => 1
{{1,2,3,4},{5,6}}
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => 1
{{1,2,3,4},{5},{6}}
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => 1

Description

The maximal size of a rise in a permutation. This is

$\max_i \sigma_{i+1}-\sigma_i$ , except for the permutations without rises, where it is

$0$ .

Matching statistic: St000845
(load all 2 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal number of elements covered by an element in a poset.

Matching statistic: St001011
(load all 4 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1,0]
 => [1,0]
 => 0
{{1,2}}
 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1
{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
{{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1
{{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
{{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
{{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
{{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
{{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
{{1,2},{3,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
{{1,4,5},{2,3}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
{{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
{{1,4},{2,3},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
{{1,5},{2,3,4}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
{{1},{2,3,4,5}}
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1},{2,3,4},{5}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
{{1,5},{2,3},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
{{1},{2,3,5},{4}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
{{1},{2,3},{4,5}}
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
{{1},{2,3},{4},{5}}
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
{{1,5},{2},{3,4}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
{{1},{2,5},{3,4}}
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
{{1},{2},{3,4,5}}
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1},{2},{3,4},{5}}
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
{{1},{2},{3},{4,5}}
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,2,3,4,5,6}}
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1
{{1,2,3,4,5},{6}}
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,2,3,4,6},{5}}
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,2,3,4},{5,6}}
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1
{{1,2,3,4},{5},{6}}
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 1

Description

Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001092

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1] => [1]
 => 0
{{1,2}}
 => [2,1] => [2,1] => [2]
 => 1
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [2,1]
 => 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1]
 => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [2,1]
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,2,3,1] => [2,1,1]
 => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => [2,1,1]
 => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,3,2,1] => [2,2]
 => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,2]
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,1]
 => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [2,2]
 => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [2,1,1]
 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [2,1,1]
 => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [2,1,1]
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,1,1,1]
 => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,2,3,1,5] => [2,1,1,1]
 => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,2,4,3,1] => [2,2,1]
 => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,2,1,5,4] => [2,2,1]
 => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,2,1,4,5] => [2,1,1,1]
 => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [5,3,2,4,1] => [2,2,1]
 => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,5,3,1,2] => [2,2,1]
 => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,3,2,1,5] => [2,2,1]
 => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
 => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
 => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
 => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,3,2,4,1] => [2,2,1]
 => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => [2,2,1]
 => 1
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,3,4,2] => [2,1,1,1]
 => 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
 => 1
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
 => 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => 1
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [5,4,3,2,1] => [2,2,1]
 => 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
 => 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
 => 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [6,2,3,4,5,1] => [2,1,1,1,1]
 => 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => [5,2,3,4,1,6] => [2,1,1,1,1]
 => 1
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [6,2,3,5,4,1] => [2,2,1,1]
 => 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [4,2,3,1,6,5] => [2,2,1,1]
 => 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [4,2,3,1,5,6] => [2,1,1,1,1]
 => 1

Description

The number of distinct even parts of a partition. See Section 3.3.1 of [1].

The following 291 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001271The competition number of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001587Half of the largest even part of an integer partition. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000729The minimal arc length of a set partition. St001075The minimal size of a block of a set partition. St000929The constant term of the character polynomial of an integer partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000629The defect of a binary word. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000054The first entry of the permutation. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000392The length of the longest run of ones in a binary word. St000456The monochromatic index of a connected graph. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000730The maximal arc length of a set partition. St000762The sum of the positions of the weak records of an integer composition. St000781The number of proper colouring schemes of a Ferrers diagram. St000823The number of unsplittable factors of the set partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000990The first ascent of a permutation. St000993The multiplicity of the largest part of an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001468The smallest fixpoint of a permutation. St001592The maximal number of simple paths between any two different vertices of a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000210Minimum over maximum difference of elements in cycles. St000297The number of leading ones in a binary word. St000317The cycle descent number of a permutation. St000352The Elizalde-Pak rank of a permutation. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000379The number of Hamiltonian cycles in a graph. St000406The number of occurrences of the pattern 3241 in a permutation. St000478Another weight of a partition according to Alladi. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000661The number of rises of length 3 of a Dyck path. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000668The least common multiple of the parts of the partition. St000699The toughness times the least common multiple of 1,. St000732The number of double deficiencies of a permutation. St000748The major index of the permutation obtained by flattening the set partition. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000842The breadth of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001062The maximal size of a block of a set partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001281The normalized isoperimetric number of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001793The difference between the clique number and the chromatic number of a graph. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000486The number of cycles of length at least 3 of a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St000258The burning number of a graph. St000918The 2-limited packing number of a graph. St000183The side length of the Durfee square of an integer partition. St000913The number of ways to refine the partition into singletons. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. 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

$eA$ . St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001175The size of a partition minus the hook length of the base cell. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001471The magnitude of a Dyck path. St001485The modular major index of a binary word. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000264The girth of a graph, which is not a tree. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001741The largest integer such that all patterns of this size are contained in the permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000058The order of a permutation. St000570The Edelman-Greene number of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St000298The order dimension or Dushnik-Miller dimension of a poset. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000065The number of entries equal to -1 in an alternating sign matrix. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000649The number of 3-excedences of a permutation. St000711The number of big exceedences of a permutation. St001434The number of negative sum pairs of a signed permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St000961The shifted major index of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000451The length of the longest pattern of the form k 1 2. St000640The rank of the largest boolean interval in a poset. St000914The sum of the values of the Möbius function of a poset. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001811The Castelnuovo-Mumford regularity of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001090The number of pop-stack-sorts needed to sort a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000028The number of stack-sorts needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000886The number of permutations with the same antidiagonal sums. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000485The length of the longest cycle of a permutation. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001260The permanent of an alternating sign matrix. St000989The number of final rises of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000007The number of saliances of the permutation. St000259The diameter of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000652The maximal difference between successive positions of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001512The minimum rank of a graph. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St000062The length of the longest increasing subsequence of the permutation. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000516The number of stretching pairs of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000822The Hadwiger number of the graph. St001093The detour number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001530The depth of a Dyck path. St001674The number of vertices of the largest induced star graph in the graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000061The number of nodes on the left branch of a binary tree. St000654The first descent of a permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. 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

$K[x]/(x^n)$ . St000461The rix statistic of a permutation. St000873The aix statistic of a permutation. St001429The number of negative entries in a signed permutation. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001890The maximum magnitude of the Möbius function of a poset. St000219The number of occurrences of the pattern 231 in a permutation. St001111The weak 2-dynamic chromatic number of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001734The lettericity of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001520The number of strict 3-descents. St001557The number of inversions of the second entry of a permutation. St001948The number of augmented double ascents of a permutation. 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

$eA$ . St001555The order of a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.