- FindStat

Your data matches 163 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001904

Mapping

St001904: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 1
[1,1] => 1
[1,2] => 2
[2,1] => 1
[1,1,1] => 1
[1,1,2] => 1
[1,2,1] => 2
[2,1,1] => 1
[1,1,3] => 1
[1,3,1] => 2
[3,1,1] => 1
[1,2,2] => 2
[2,1,2] => 1
[2,2,1] => 1
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 1
[3,2,1] => 1
[1,1,1,1] => 1
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,2,1,1] => 2
[2,1,1,1] => 1
[1,1,1,3] => 1
[1,1,3,1] => 1
[1,3,1,1] => 2
[3,1,1,1] => 1
[1,1,1,4] => 1
[1,1,4,1] => 1
[1,4,1,1] => 2
[4,1,1,1] => 1
[1,1,2,2] => 1
[1,2,1,2] => 2
[1,2,2,1] => 2
[2,1,1,2] => 1
[2,1,2,1] => 1
[2,2,1,1] => 1
[1,1,2,3] => 1
[1,1,3,2] => 1
[1,2,1,3] => 2
[1,2,3,1] => 3
[1,3,1,2] => 2
[1,3,2,1] => 2
[2,1,1,3] => 1
[2,1,3,1] => 1
[2,3,1,1] => 2
[3,1,1,2] => 1
[3,1,2,1] => 1

Description

The length of the initial strictly increasing segment of a parking function.

Matching statistic: St000382
(load all 2 compositions to match this statistic)

Mapping

Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1
[1,1] => [1,1] => 1
[1,2] => [1,1] => 1
[2,1] => [2] => 2
[1,1,1] => [1,1,1] => 1
[1,1,2] => [1,1,1] => 1
[1,2,1] => [1,2] => 1
[2,1,1] => [2,1] => 2
[1,1,3] => [1,2] => 1
[1,3,1] => [1,1,1] => 1
[3,1,1] => [2,1] => 2
[1,2,2] => [1,1,1] => 1
[2,1,2] => [2,1] => 2
[2,2,1] => [1,2] => 1
[1,2,3] => [1,1,1] => 1
[1,3,2] => [1,2] => 1
[2,1,3] => [2,1] => 2
[2,3,1] => [1,2] => 1
[3,1,2] => [2,1] => 2
[3,2,1] => [3] => 3
[1,1,1,1] => [1,1,1,1] => 1
[1,1,1,2] => [1,1,1,1] => 1
[1,1,2,1] => [1,1,2] => 1
[1,2,1,1] => [1,2,1] => 1
[2,1,1,1] => [2,1,1] => 2
[1,1,1,3] => [1,1,2] => 1
[1,1,3,1] => [1,1,1,1] => 1
[1,3,1,1] => [1,2,1] => 1
[3,1,1,1] => [2,1,1] => 2
[1,1,1,4] => [1,1,2] => 1
[1,1,4,1] => [1,2,1] => 1
[1,4,1,1] => [1,1,1,1] => 1
[4,1,1,1] => [2,1,1] => 2
[1,1,2,2] => [1,1,1,1] => 1
[1,2,1,2] => [1,2,1] => 1
[1,2,2,1] => [1,1,2] => 1
[2,1,1,2] => [2,1,1] => 2
[2,1,2,1] => [2,2] => 2
[2,2,1,1] => [1,2,1] => 1
[1,1,2,3] => [1,1,1,1] => 1
[1,1,3,2] => [1,1,2] => 1
[1,2,1,3] => [1,2,1] => 1
[1,2,3,1] => [1,1,2] => 1
[1,3,1,2] => [1,2,1] => 1
[1,3,2,1] => [1,3] => 1
[2,1,1,3] => [2,1,1] => 2
[2,1,3,1] => [2,2] => 2
[2,3,1,1] => [1,2,1] => 1
[3,1,1,2] => [2,1,1] => 2
[3,1,2,1] => [2,2] => 2

Description

The first part of an integer composition.

Matching statistic: St000383
(load all 5 compositions to match this statistic)

Mapping

Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1
[1,1] => [1,1] => 1
[1,2] => [1,1] => 1
[2,1] => [2] => 2
[1,1,1] => [1,1,1] => 1
[1,1,2] => [1,1,1] => 1
[1,2,1] => [1,2] => 2
[2,1,1] => [2,1] => 1
[1,1,3] => [1,2] => 2
[1,3,1] => [1,1,1] => 1
[3,1,1] => [2,1] => 1
[1,2,2] => [1,1,1] => 1
[2,1,2] => [2,1] => 1
[2,2,1] => [1,2] => 2
[1,2,3] => [1,1,1] => 1
[1,3,2] => [1,2] => 2
[2,1,3] => [2,1] => 1
[2,3,1] => [1,2] => 2
[3,1,2] => [2,1] => 1
[3,2,1] => [3] => 3
[1,1,1,1] => [1,1,1,1] => 1
[1,1,1,2] => [1,1,1,1] => 1
[1,1,2,1] => [1,1,2] => 2
[1,2,1,1] => [1,2,1] => 1
[2,1,1,1] => [2,1,1] => 1
[1,1,1,3] => [1,1,2] => 2
[1,1,3,1] => [1,1,1,1] => 1
[1,3,1,1] => [1,2,1] => 1
[3,1,1,1] => [2,1,1] => 1
[1,1,1,4] => [1,1,2] => 2
[1,1,4,1] => [1,2,1] => 1
[1,4,1,1] => [1,1,1,1] => 1
[4,1,1,1] => [2,1,1] => 1
[1,1,2,2] => [1,1,1,1] => 1
[1,2,1,2] => [1,2,1] => 1
[1,2,2,1] => [1,1,2] => 2
[2,1,1,2] => [2,1,1] => 1
[2,1,2,1] => [2,2] => 2
[2,2,1,1] => [1,2,1] => 1
[1,1,2,3] => [1,1,1,1] => 1
[1,1,3,2] => [1,1,2] => 2
[1,2,1,3] => [1,2,1] => 1
[1,2,3,1] => [1,1,2] => 2
[1,3,1,2] => [1,2,1] => 1
[1,3,2,1] => [1,3] => 3
[2,1,1,3] => [2,1,1] => 1
[2,1,3,1] => [2,2] => 2
[2,3,1,1] => [1,2,1] => 1
[3,1,1,2] => [2,1,1] => 1
[3,1,2,1] => [2,2] => 2

Description

The last part of an integer composition.

Matching statistic: St000214
(load all 11 compositions to match this statistic)

Mapping

Mp00053: Parking functions —to car permutation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 0 = 1 - 1
[1,1] => [1,2] => 0 = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => 1 = 2 - 1
[1,1,1] => [1,2,3] => 0 = 1 - 1
[1,1,2] => [1,2,3] => 0 = 1 - 1
[1,2,1] => [1,2,3] => 0 = 1 - 1
[2,1,1] => [2,1,3] => 1 = 2 - 1
[1,1,3] => [1,2,3] => 0 = 1 - 1
[1,3,1] => [1,3,2] => 1 = 2 - 1
[3,1,1] => [2,3,1] => 0 = 1 - 1
[1,2,2] => [1,2,3] => 0 = 1 - 1
[2,1,2] => [2,1,3] => 1 = 2 - 1
[2,2,1] => [3,1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3] => [2,1,3] => 1 = 2 - 1
[2,3,1] => [3,1,2] => 0 = 1 - 1
[3,1,2] => [2,3,1] => 0 = 1 - 1
[3,2,1] => [3,2,1] => 2 = 3 - 1
[1,1,1,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,2] => [1,2,3,4] => 0 = 1 - 1
[1,1,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,2,1,1] => [1,2,3,4] => 0 = 1 - 1
[2,1,1,1] => [2,1,3,4] => 1 = 2 - 1
[1,1,1,3] => [1,2,3,4] => 0 = 1 - 1
[1,1,3,1] => [1,2,3,4] => 0 = 1 - 1
[1,3,1,1] => [1,3,2,4] => 1 = 2 - 1
[3,1,1,1] => [2,3,1,4] => 0 = 1 - 1
[1,1,1,4] => [1,2,3,4] => 0 = 1 - 1
[1,1,4,1] => [1,2,4,3] => 1 = 2 - 1
[1,4,1,1] => [1,3,4,2] => 0 = 1 - 1
[4,1,1,1] => [2,3,4,1] => 0 = 1 - 1
[1,1,2,2] => [1,2,3,4] => 0 = 1 - 1
[1,2,1,2] => [1,2,3,4] => 0 = 1 - 1
[1,2,2,1] => [1,2,3,4] => 0 = 1 - 1
[2,1,1,2] => [2,1,3,4] => 1 = 2 - 1
[2,1,2,1] => [2,1,3,4] => 1 = 2 - 1
[2,2,1,1] => [3,1,2,4] => 0 = 1 - 1
[1,1,2,3] => [1,2,3,4] => 0 = 1 - 1
[1,1,3,2] => [1,2,3,4] => 0 = 1 - 1
[1,2,1,3] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,1] => [1,2,3,4] => 0 = 1 - 1
[1,3,1,2] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,1] => [1,3,2,4] => 1 = 2 - 1
[2,1,1,3] => [2,1,3,4] => 1 = 2 - 1
[2,1,3,1] => [2,1,3,4] => 1 = 2 - 1
[2,3,1,1] => [3,1,2,4] => 0 = 1 - 1
[3,1,1,2] => [2,3,1,4] => 0 = 1 - 1
[3,1,2,1] => [2,3,1,4] => 0 = 1 - 1

Description

The number of adjacencies of a permutation. An adjacency of a permutation

$\pi$ is an index

$i$ such that

$\pi(i)-1 = \pi(i+1)$ . Adjacencies are also known as ''small descents''. This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].

Matching statistic: St000025
(load all 3 compositions to match this statistic)

Mapping

Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1,0]
 => 1
[1,1] => [1,1] => [1,0,1,0]
 => 1
[1,2] => [1,1] => [1,0,1,0]
 => 1
[2,1] => [2] => [1,1,0,0]
 => 2
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => 1
[1,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 1
[1,2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,1] => [2,1] => [1,1,0,0,1,0]
 => 2
[1,1,3] => [1,2] => [1,0,1,1,0,0]
 => 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 1
[3,1,1] => [2,1] => [1,1,0,0,1,0]
 => 2
[1,2,2] => [1,1,1] => [1,0,1,0,1,0]
 => 1
[2,1,2] => [2,1] => [1,1,0,0,1,0]
 => 2
[2,2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
 => 1
[1,3,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => [2,1] => [1,1,0,0,1,0]
 => 2
[2,3,1] => [1,2] => [1,0,1,1,0,0]
 => 1
[3,1,2] => [2,1] => [1,1,0,0,1,0]
 => 2
[3,2,1] => [3] => [1,1,1,0,0,0]
 => 3
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,1,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,1,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[2,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,1,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,3,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,1,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,1,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,4,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[4,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,1,2,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,2,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,2,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[2,1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[2,1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,1,2,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,1,3,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,2,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,2,3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,3,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,3,2,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[2,1,3,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,3,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[3,1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2

Description

The number of initial rises of a Dyck path. In other words, this is the height of the first peak of

$D$ .

Matching statistic: St000026
(load all 2 compositions to match this statistic)

Mapping

Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1,0]
 => 1
[1,1] => [1,1] => [1,0,1,0]
 => 1
[1,2] => [1,1] => [1,0,1,0]
 => 1
[2,1] => [2] => [1,1,0,0]
 => 2
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => 1
[1,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 1
[1,2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,1] => [2,1] => [1,1,0,0,1,0]
 => 2
[1,1,3] => [1,2] => [1,0,1,1,0,0]
 => 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 1
[3,1,1] => [2,1] => [1,1,0,0,1,0]
 => 2
[1,2,2] => [1,1,1] => [1,0,1,0,1,0]
 => 1
[2,1,2] => [2,1] => [1,1,0,0,1,0]
 => 2
[2,2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
 => 1
[1,3,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => [2,1] => [1,1,0,0,1,0]
 => 2
[2,3,1] => [1,2] => [1,0,1,1,0,0]
 => 1
[3,1,2] => [2,1] => [1,1,0,0,1,0]
 => 2
[3,2,1] => [3] => [1,1,1,0,0,0]
 => 3
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,1,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,1,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[2,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,1,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,3,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,1,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,1,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,4,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[4,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,1,2,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,2,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,2,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[2,1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[2,1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,1,2,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,1,3,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,2,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,2,3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,3,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,3,2,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[2,1,3,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,3,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[3,1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2

Description

The position of the first return of a Dyck path.

Matching statistic: St000273
(load all 3 compositions to match this statistic)

Mapping

Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The domination number of a graph. The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.

Matching statistic: St000287
(load all 3 compositions to match this statistic)

Mapping

Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000287: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of connected components of a graph.

Matching statistic: St000544
(load all 3 compositions to match this statistic)

Mapping

Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000544: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cop number of a graph. This is the minimal number of cops needed to catch the robber. The algorithm is from [2].

Matching statistic: St000745
(load all 11 compositions to match this statistic)

Mapping

Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [[1]]
 => 1
[1,1] => [1,2] => [[1,2]]
 => 1
[1,2] => [1,2] => [[1,2]]
 => 1
[2,1] => [2,1] => [[1],[2]]
 => 2
[1,1,1] => [1,2,3] => [[1,2,3]]
 => 1
[1,1,2] => [1,2,3] => [[1,2,3]]
 => 1
[1,2,1] => [1,3,2] => [[1,2],[3]]
 => 1
[2,1,1] => [2,3,1] => [[1,3],[2]]
 => 2
[1,1,3] => [1,2,3] => [[1,2,3]]
 => 1
[1,3,1] => [1,3,2] => [[1,2],[3]]
 => 1
[3,1,1] => [2,3,1] => [[1,3],[2]]
 => 2
[1,2,2] => [1,2,3] => [[1,2,3]]
 => 1
[2,1,2] => [2,1,3] => [[1,3],[2]]
 => 2
[2,2,1] => [3,1,2] => [[1,2],[3]]
 => 1
[1,2,3] => [1,2,3] => [[1,2,3]]
 => 1
[1,3,2] => [1,3,2] => [[1,2],[3]]
 => 1
[2,1,3] => [2,1,3] => [[1,3],[2]]
 => 2
[2,3,1] => [3,1,2] => [[1,2],[3]]
 => 1
[3,1,2] => [2,3,1] => [[1,3],[2]]
 => 2
[3,2,1] => [3,2,1] => [[1],[2],[3]]
 => 3
[1,1,1,1] => [1,2,3,4] => [[1,2,3,4]]
 => 1
[1,1,1,2] => [1,2,3,4] => [[1,2,3,4]]
 => 1
[1,1,2,1] => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[1,2,1,1] => [1,3,4,2] => [[1,2,4],[3]]
 => 1
[2,1,1,1] => [2,3,4,1] => [[1,3,4],[2]]
 => 2
[1,1,1,3] => [1,2,3,4] => [[1,2,3,4]]
 => 1
[1,1,3,1] => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[1,3,1,1] => [1,3,4,2] => [[1,2,4],[3]]
 => 1
[3,1,1,1] => [2,3,4,1] => [[1,3,4],[2]]
 => 2
[1,1,1,4] => [1,2,3,4] => [[1,2,3,4]]
 => 1
[1,1,4,1] => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[1,4,1,1] => [1,3,4,2] => [[1,2,4],[3]]
 => 1
[4,1,1,1] => [2,3,4,1] => [[1,3,4],[2]]
 => 2
[1,1,2,2] => [1,2,3,4] => [[1,2,3,4]]
 => 1
[1,2,1,2] => [1,3,2,4] => [[1,2,4],[3]]
 => 1
[1,2,2,1] => [1,4,2,3] => [[1,2,3],[4]]
 => 1
[2,1,1,2] => [2,3,1,4] => [[1,3,4],[2]]
 => 2
[2,1,2,1] => [2,4,1,3] => [[1,3],[2,4]]
 => 2
[2,2,1,1] => [3,4,1,2] => [[1,2],[3,4]]
 => 1
[1,1,2,3] => [1,2,3,4] => [[1,2,3,4]]
 => 1
[1,1,3,2] => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[1,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
 => 1
[1,2,3,1] => [1,4,2,3] => [[1,2,3],[4]]
 => 1
[1,3,1,2] => [1,3,4,2] => [[1,2,4],[3]]
 => 1
[1,3,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
 => 1
[2,1,1,3] => [2,3,1,4] => [[1,3,4],[2]]
 => 2
[2,1,3,1] => [2,4,1,3] => [[1,3],[2,4]]
 => 2
[2,3,1,1] => [3,4,1,2] => [[1,2],[3,4]]
 => 1
[3,1,1,2] => [2,3,4,1] => [[1,3,4],[2]]
 => 2
[3,1,2,1] => [2,4,3,1] => [[1,3],[2],[4]]
 => 2

Description

The index of the last row whose first entry is the row number in a standard Young tableau.

The following 153 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000916The packing number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001829The common independence number of a graph. St000237The number of small exceedances. St000439The position of the first down step of a Dyck path. St000441The number of successions of a permutation. St000007The number of saliances of the permutation. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000054The first entry of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000286The number of connected components of the complement of a graph. St000297The number of leading ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000363The number of minimal vertex covers of a graph. St000501The size of the first part in the decomposition of a permutation. St000505The biggest entry in the block containing the 1. St000542The number of left-to-right-minima of a permutation. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000654The first descent of a permutation. St000740The last entry of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in 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. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001316The domatic number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001937The size of the center of a parking function. St000051The size of the left subtree of a binary tree. St000090The variation of a composition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000338The number of pixed points of a permutation. St000546The number of global descents of a permutation. St000738The first entry in the last row of a standard tableau. St001176The size of a partition minus its first part. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. 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. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in 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. St001631The number of simple modules

$S$ with

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

$A$ of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000990The first ascent of a permutation. St000061The number of nodes on the left branch of a binary tree. St000504The cardinality of the first block of a set partition. St000678The number of up steps after the last double rise of a Dyck path. St000823The number of unsplittable factors of the set partition. St000502The number of successions of a set partitions. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St000993The multiplicity of the largest part of an integer partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001568The smallest positive integer that does not appear twice in the partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001432The order dimension of the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001571The Cartan determinant of the integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000937The number of positive values of the symmetric group character corresponding to the partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St001118The acyclic chromatic index of a graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001378The product of the cohook lengths of the integer partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001933The largest multiplicity of a part in an integer partition. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St001128The exponens consonantiae of a partition. St000454The largest eigenvalue of a graph if it is integral. St000260The radius of a connected graph. St000177The number of free tiles in the pattern. St000178Number of free entries. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001060The distinguishing index of a graph. 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. St001330The hat guessing number of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.