- FindStat

Your data matches 366 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000228
(load all 41 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a partition. This statistic is the constant statistic of the level sets.

Matching statistic: St000384
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal part of the shifted composition of an integer partition. A partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding

$i-1$ to the

$i$ -th part. The statistic is then

$\operatorname{max}_i\{ \lambda_i + i - 1 \}$ . See also [[St000380]].

Matching statistic: St000459
(load all 12 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.

Matching statistic: St000460
(load all 12 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The hook length of the last cell along the main diagonal of an integer partition.

Matching statistic: St000531
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000531: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The leading coefficient of the rook polynomial of an integer partition. Let

$m$ be the minimum of the number of parts and the size of the first part of an integer partition

$\lambda$ . Then this statistic yields the number of ways to place

$m$ non-attacking rooks on the Ferrers board of

$\lambda$ .

Matching statistic: St000784
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximum of the length and the largest part of the integer partition. This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1]. See also [[St001214]].

Matching statistic: St000870
(load all 12 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The product of the hook lengths of the diagonal cells in an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells

$(i,i)$ of a partition.

Matching statistic: St001360
(load all 3 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001360: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of covering relations in Young's lattice below a partition.

Matching statistic: St001659
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001659: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ways to place as many non-attacking rooks as possible on a Ferrers board.

Matching statistic: St000063
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of linear extensions of a certain poset defined for an integer partition. The poset is constructed in David Speyer's answer to Matt Fayers' question [3]. The value at the partition

$\lambda$ also counts cover-inclusive Dyck tilings of

$\lambda\setminus\mu$ , summed over all

$\mu$ , as noticed by Philippe Nadeau in a comment. This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.

The following 356 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000108The number of partitions contained in the given partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St000681The Grundy value of Chomp on Ferrers diagrams. St000806The semiperimeter of the associated bargraph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001400The total number of Littlewood-Richardson tableaux of given shape. St000013The height of a Dyck path. St000293The number of inversions of a binary word. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000469The distinguishing number of a graph. St000519The largest length of a factor maximising the subword complexity. St000636The hull number of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000922The minimal number such that all substrings of this length are unique. St000926The clique-coclique number of a graph. St000982The length of the longest constant subword. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001267The length of the Lyndon factorization of the binary word. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001437The flex of a binary word. St001523The degree of symmetry of a Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000744The length of the path to the largest entry in a standard Young tableau. St000778The metric dimension of a graph. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000921The number of internal inversions of a binary word. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001658The total number of rook placements on a Ferrers board. St001949The rigidity index of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St000011The number of touch points (or returns) of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000026The position of the first return of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000153The number of adjacent cycles of a permutation. St000203The number of external nodes of a binary tree. St000246The number of non-inversions of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000277The number of ribbon shaped standard tableaux. St000290The major index of a binary word. St000294The number of distinct factors of a binary word. St000296The length of the symmetric border of a binary word. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000451The length of the longest pattern of the form k 1 2. St000518The number of distinct subsequences in a binary word. St000528The height of a poset. St000548The number of different non-empty partial sums of an integer partition. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000628The balance of a binary word. St000651The maximal size of a rise in a permutation. St000657The smallest part of an integer composition. St000676The number of odd rises of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000696The number of cycles in the breakpoint graph of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000734The last entry in the first row of a standard tableau. St000808The number of up steps of the associated bargraph. St000825The sum of the major and the inverse major index of a permutation. St000883The number of longest increasing subsequences of a permutation. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000983The length of the longest alternating subword. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St001090The number of pop-stack-sorts needed to sort a permutation. St001249Sum of the odd parts of a partition. St001342The number of vertices in the center of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001372The length of a longest cyclic run of ones of a binary word. 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. St001485The modular major index of a binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001554The number of distinct nonempty subtrees of a binary tree. St001733The number of weak left to right maxima of a Dyck path. St001746The coalition number of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000022The number of fixed points of a permutation. St000067The inversion number of the alternating sign matrix. St000070The number of antichains in a poset. St000085The number of linear extensions of the tree. St000110The number of permutations less than or equal to a permutation in left weak order. St000141The maximum drop size of a permutation. St000163The size of the orbit of the set partition under rotation. St000171The degree of the graph. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000245The number of ascents of a permutation. St000248The number of anti-singletons of a set partition. St000288The number of ones in a binary word. St000306The bounce count of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000336The leg major index of a standard tableau. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000385The number of vertices with out-degree 1 in a binary tree. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000439The position of the first down step of a Dyck path. St000441The number of successions of a permutation. St000445The number of rises of length 1 of a Dyck path. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000627The exponent of a binary word. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000674The number of hills of a Dyck path. St000691The number of changes of a binary word. St000703The number of deficiencies of a permutation. St000728The dimension of a set partition. St000819The propagating number of a perfect matching. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000932The number of occurrences of the pattern UDU in a Dyck path. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001312Number of parabolic noncrossing partitions indexed by the composition. St001313The number of Dyck paths above the lattice path given by a binary word. 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. St001631The number of simple modules

$S$ with

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

$A$ of the poset. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000878The number of ones minus the number of zeros of a binary word. St001688The sum of the squares of the heights of the peaks of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000444The length of the maximal rise of a Dyck path. St000625The sum of the minimal distances to a greater element. St000678The number of up steps after the last double rise of a Dyck path. St000722The number of different neighbourhoods in a graph. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001074The number of inversions of the cyclic embedding of a permutation. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001566The length of the longest arithmetic progression in a permutation. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000420The number of Dyck paths that are weakly above a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000520The number of patterns in a permutation. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001405The number of bonds in a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000237The number of small exceedances. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000731The number of double exceedences of a permutation. St001645The pebbling number of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000438The position of the last up step in a Dyck path. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000874The position of the last double rise in a Dyck path. St000993The multiplicity of the largest part of an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001933The largest multiplicity of a part in an integer partition. St001176The size of a partition minus its first part. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001330The hat guessing number of a graph. St000947The major index east count of a Dyck path. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000377The dinv defect of an integer partition. St001725The harmonious chromatic number of a graph. St000271The chromatic index of a graph. St000050The depth or height of a binary tree. St000946The sum of the skew hook positions in a Dyck path. St001439The number of even weak deficiencies and of odd weak exceedences. St000416The number of inequivalent increasing trees of an ordered tree. St000521The number of distinct subtrees of an ordered tree. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001958The degree of the polynomial interpolating the values of a permutation. St000837The number of ascents of distance 2 of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000087The number of induced subgraphs. St000144The pyramid weight of the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000024The number of double up and double down steps of a Dyck path. St000529The number of permutations whose descent word is the given binary word. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000025The number of initial rises of a Dyck path. St000058The order of a permutation. St000064The number of one-box pattern of a permutation. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000189The number of elements in the poset. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000235The number of indices that are not cyclical small weak excedances. St000236The number of cyclical small weak excedances. St000240The number of indices that are not small excedances. St000308The height of the tree associated to a permutation. St000501The size of the first part in the decomposition of a permutation. St000656The number of cuts of a poset. St000673The number of non-fixed points of a permutation. St000680The Grundy value for Hackendot on posets. St000686The finitistic dominant dimension of a Dyck path. St000717The number of ordinal summands of a poset. St000844The size of the largest block in the direct sum decomposition of a permutation. St000906The length of the shortest maximal chain in a poset. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001161The major index north count of a Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001717The largest size of an interval in a poset. St001809The index of the step at the first peak of maximal height in a Dyck path. St000030The sum of the descent differences of a permutations. St000053The number of valleys of the Dyck path. St000060The greater neighbor of the maximum. St000080The rank of the poset. St000089The absolute variation of a composition. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St000619The number of cyclic descents of a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000969We 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

$c_{n-1}$ . St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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. St001246The maximal difference between two consecutive entries of a permutation. St001468The smallest fixpoint of a permutation. St001480The number of simple summands of the module J^2/J^3. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001516The number of cyclic bonds of a permutation. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000242The number of indices that are not cyclical small weak excedances. St000331The number of upper interactions of a Dyck path. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. 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. 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001910The height of the middle non-run of a Dyck path. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St001883The mutual visibility number of a graph. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000530The number of permutations with the same descent word as the given permutation. St000702The number of weak deficiencies of a permutation. St000890The number of nonzero entries in an alternating sign matrix. St000924The number of topologically connected components of a perfect matching. St001461The number of topologically connected components of the chord diagram of a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001082The number of boxed occurrences of 123 in a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001911A descent variant minus the number of inversions. St001925The minimal number of zeros in a row of an alternating sign matrix. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St001130The number of two successive successions in a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001959The product of the heights of the peaks of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St000225Difference between largest and smallest parts in a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001280The number of parts of an integer partition that are at least two. St000454The largest eigenvalue of a graph if it is integral. St000005The bounce statistic of a Dyck path.