- FindStat

Your data matches 619 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001175
(load all 3 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.

Matching statistic: St000017
(load all 4 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inversions of a standard tableau. Let

$T$ be a tableau. An inversion is an attacking pair

$(c,d)$ of the shape of

$T$ (see [[St000016]] for a definition of this) such that the entry of

$c$ in

$T$ is greater than the entry of

$d$ .

Matching statistic: St000142
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of even parts of a partition.

Matching statistic: St000473
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts of a partition that are strictly bigger than the number of ones. This is part of the definition of Dyson's crank of a partition, see [[St000474]].

Matching statistic: St001033
(load all 3 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The normalized area of the parallelogram polyomino associated with the Dyck path. The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path. The area itself is equidistributed with [[St001034]] and with [[St000395]].

Matching statistic: St001251
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St001252
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Half the sum of the even parts of a partition.

Matching statistic: St001596
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of two-by-two squares inside a skew partition. This is, the number of cells

$(i,j)$ in a skew partition for which the box

$(i+1,j+1)$ is also a cell inside the skew partition.

Matching statistic: St000321
(load all 3 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of integer partitions of n that are dominated by an integer partition. A partition

$\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition

$\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if

$\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all

$k$ .

Matching statistic: St000345
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of refinements of a partition. A partition

$\lambda$ refines a partition

$\mu$ if the parts of

$\mu$ can be subdivided to obtain the parts of

$\lambda$ .

The following 609 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000935The number of ordered refinements of an integer partition. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001256Number of simple reflexive modules that are 2-stable reflexive. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000649The number of 3-excedences of a permutation. St001513The number of nested exceedences of a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001933The largest multiplicity of a part in an integer partition. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000257The number of distinct parts of a partition that occur at least twice. St000481The number of upper covers of a partition in dominance order. St000048The multinomial of the parts of a partition. St000346The number of coarsenings of a partition. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000480The number of lower covers of a partition in dominance order. St001092The number of distinct even parts of a partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001793The difference between the clique number and the chromatic number of a graph. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St000323The minimal crossing number of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000403The Szeged index minus the Wiener index of a graph. St000448The number of pairs of vertices of a graph with distance 2. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. 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. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001708The number of pairs of vertices of different degree in a graph. St001764The number of non-convex subsets of vertices in a graph. St001797The number of overfull subgraphs of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001799The number of proper separations of a graph. St000273The domination number of a graph. St000349The number of different adjacency matrices of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000544The cop number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001829The common independence number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000287The number of connected components of a graph. St001282The number of graphs with the same chromatic polynomial. St001518The number of graphs with the same ordinary spectrum as the given graph. St000095The number of triangles of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000322The skewness of a graph. St001306The number of induced paths on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. 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. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. 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. St001742The difference of the maximal and the minimal degree in a graph. St001871The number of triconnected components of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000286The number of connected components of the complement of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001272The number of graphs with the same degree sequence. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St001765The number of connected components of the friends and strangers graph. St001917The order of toric promotion on the set of labellings of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000931The number of occurrences of the pattern UUU in a Dyck path. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000759The smallest missing part in an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000264The girth of a graph, which is not a tree. St000296The length of the symmetric border of a binary word. St000629The defect of a binary word. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001371The length of the longest Yamanouchi prefix of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St001570The minimal number of edges to add to make a graph Hamiltonian. St001172The number of 1-rises at odd height of a Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001593This is the number of standard Young tableaux of the given shifted shape. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000022The number of fixed points of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000214The number of adjacencies of a permutation. St000223The number of nestings in the permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000352The Elizalde-Pak rank of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000007The number of saliances of the permutation. St001280The number of parts of an integer partition that are at least two. St001657The number of twos in an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St001389The number of partitions of the same length below the given integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001367The smallest number which does not occur as degree of a vertex in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001060The distinguishing index of a graph. St000225Difference between largest and smallest parts in a partition. St001214The aft of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000783The side length of the largest staircase partition fitting into a partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000310The minimal degree of a vertex of a graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000379The number of Hamiltonian cycles in a graph. St000456The monochromatic index of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001890The maximum magnitude of the Möbius function of a poset. St001645The pebbling number of a connected graph. St001571The Cartan determinant of the integer partition. St000993The multiplicity of the largest part of an integer partition. St001696The natural major index of a standard Young tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000003The number of standard Young tableaux of the partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000897The number of different multiplicities of parts of an integer partition. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001432The order dimension of the partition. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. 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. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. 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. St000260The radius of a connected graph. St000259The diameter of a connected graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000052The number of valleys of a Dyck path not on the x-axis. St000292The number of ascents of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000944The 3-degree of an integer partition. St000288The number of ones in a binary word. St000290The major index of a binary word. St000291The number of descents of a binary word. St000297The number of leading ones in a binary word. St000306The bounce count of a Dyck path. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000627The exponent of a binary word. St000628The balance of a binary word. St000655The length of the minimal rise of a Dyck path. St000792The Grundy value for the game of ruler on a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000920The logarithmic height of a Dyck path. St001372The length of a longest cyclic run of ones of a binary word. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000630The length of the shortest palindromic decomposition of a binary word. St001176The size of a partition minus its first part. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St000699The toughness times the least common multiple of 1,. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001281The normalized isoperimetric number of a graph. St001118The acyclic chromatic index of a graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St000315The number of isolated vertices of a graph. St000879The number of long braid edges in the graph of braid moves of a permutation. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St000258The burning number of a graph. St000918The 2-limited packing number of a graph. St001271The competition number of a graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001964The interval resolution global dimension of a poset. St001260The permanent of an alternating sign matrix. St001479The number of bridges of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000693The modular (standard) major index of a standard tableau. St000735The last entry on the main diagonal of a standard tableau. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001141The number of occurrences of hills of size 3 in a Dyck path. St000787The number of flips required to make a perfect matching noncrossing. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St000657The smallest part of an integer composition. St000788The number of nesting-similar perfect matchings of a perfect matching. St001316The domatic number of a graph. 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

$eA$ . St000383The last part of an integer composition. 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. St000012The area of a Dyck path. St000295The length of the border of a binary word. St000340The number of non-final maximal constant sub-paths of length greater than one. St000348The non-inversion sum of a binary word. St000386The number of factors DDU in a Dyck path. St000478Another weight of a partition according to Alladi. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000660The number of rises of length at least 3 of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000979Half of MacMahon's equal index of a Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001695The natural comajor index of a standard Young tableau. St001697The shifted 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. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000391The sum of the positions of the ones in a binary word. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000691The number of changes of a binary word. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000753The Grundy value for the game of Kayles on a binary word. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000847The number of standard Young tableaux whose descent set is the binary word. St000933The number of multipartitions of sizes given by an integer partition. St001313The number of Dyck paths above the lattice path given by a binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001485The modular major index of a binary word. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001732The number of peaks visible from the left. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001809The index of the step at the first peak of maximal height in a Dyck path. St001884The number of borders of a binary word. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000439The position of the first down step of a Dyck path. St000983The length of the longest alternating subword. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. 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. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000977MacMahon's equal index of a Dyck path. St000978The sum of the positions of double down-steps of a Dyck path. St000984The number of boxes below precisely one peak. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001621The number of atoms of a lattice. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001301The first Betti number of the order complex associated with the poset. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000024The number of double up and double down steps of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000075The orbit size of a standard tableau under promotion. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001196The global dimension of

$A$ minus the global dimension of

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

$eA$ . St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001498The normalised height of a Nakayama algebra with magnitude 1. St001827The number of two-component spanning forests of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000351The determinant of the adjacency matrix of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000948The chromatic discriminant of a graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001119The length of a shortest maximal path in a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001341The number of edges in the center of a graph. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001691The number of kings in a graph. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000117The number of centered tunnels of a Dyck path. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000406The number of occurrences of the pattern 3241 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000954Number of times the corresponding LNakayama algebra has

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

$i>0$ . St000989The number of final rises of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. 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. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000331The number of upper interactions of a Dyck path. St000461The rix statistic of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000535The rank-width of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000694The number of affine bounded permutations that project to a given permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001081The number of minimal length factorizations of a permutation into star transpositions. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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. St001191Number of simple modules

$S$ with

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

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

$A$ with global dimension

$g$ . St001192The maximal dimension of

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

$S$ over the corresponding Nakayama algebra

$A$ . St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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

$K[x]/(x^n)$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. 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. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001512The minimum rank of a graph. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000006The dinv of a Dyck path. St000015The number of peaks of a Dyck path. St000299The number of nonisomorphic vertex-induced subtrees. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000542The number of left-to-right-minima of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000955Number of times one has

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

$i>0$ for the corresponding LNakayama algebra. St001093The detour number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -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

$eA$ . St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. 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

$eA$ . St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001299The product of all non-zero projective dimensions of simple modules of 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. 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. St001488The number of corners of a skew partition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St001674The number of vertices of the largest induced star graph in the graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000268The number of strongly connected orientations of a graph. St000344The number of strongly connected outdegree sequences of a graph. St001073The number of nowhere zero 3-flows of a graph. St001826The maximal number of leaves on a vertex of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St000917The open packing number of a graph. St001672The restrained domination number of a graph. St001111The weak 2-dynamic chromatic number of a graph.