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Your data matches 733 different statistics following compositions of up to 3 maps.
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Matching statistic: St000686
(load all 43 compositions to match this statistic)

Mapping

St000686: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => 1 = 2 - 1

Description

The finitistic dominant dimension of a Dyck path. To every LNakayama algebra there is a corresponding Dyck path, see also [[St000684]]. We associate the finitistic dominant dimension of the algebra to the corresponding Dyck path.

Matching statistic: St001239
(load all 32 compositions to match this statistic)

Mapping

St001239: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
 => 1 = 2 - 1

Description

The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.

Matching statistic: St001530
(load all 41 compositions to match this statistic)

Mapping

St001530: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => 1 = 2 - 1

Description

The depth of a Dyck path. That is the depth of the corresponding Nakayama algebra with a linear quiver.

Matching statistic: St001192
(load all 29 compositions to match this statistic)

Mapping

St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => 0 = 2 - 2
[1,0,1,0]
 => 1 = 3 - 2
[1,1,0,0]
 => 0 = 2 - 2
[1,0,1,0,1,0]
 => 1 = 3 - 2
[1,0,1,1,0,0]
 => 1 = 3 - 2
[1,1,0,0,1,0]
 => 1 = 3 - 2
[1,1,0,1,0,0]
 => 2 = 4 - 2
[1,1,1,0,0,0]
 => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
 => 1 = 3 - 2
[1,0,1,0,1,1,0,0]
 => 1 = 3 - 2
[1,0,1,1,0,0,1,0]
 => 1 = 3 - 2
[1,0,1,1,0,1,0,0]
 => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
 => 1 = 3 - 2
[1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
 => 1 = 3 - 2
[1,1,0,1,0,0,1,0]
 => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
 => 2 = 4 - 2
[1,1,0,1,1,0,0,0]
 => 2 = 4 - 2
[1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
[1,1,1,0,0,1,0,0]
 => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
 => 3 = 5 - 2
[1,1,1,1,0,0,0,0]
 => 0 = 2 - 2

Description

The maximal dimension of

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

$S$ over the corresponding Nakayama algebra

$A$ .

Matching statistic: St001294
(load all 43 compositions to match this statistic)

Mapping

St001294: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => 0 = 2 - 2
[1,0,1,0]
 => 1 = 3 - 2
[1,1,0,0]
 => 0 = 2 - 2
[1,0,1,0,1,0]
 => 2 = 4 - 2
[1,0,1,1,0,0]
 => 1 = 3 - 2
[1,1,0,0,1,0]
 => 1 = 3 - 2
[1,1,0,1,0,0]
 => 1 = 3 - 2
[1,1,1,0,0,0]
 => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
 => 3 = 5 - 2
[1,0,1,0,1,1,0,0]
 => 2 = 4 - 2
[1,0,1,1,0,0,1,0]
 => 1 = 3 - 2
[1,0,1,1,0,1,0,0]
 => 2 = 4 - 2
[1,0,1,1,1,0,0,0]
 => 1 = 3 - 2
[1,1,0,0,1,0,1,0]
 => 2 = 4 - 2
[1,1,0,0,1,1,0,0]
 => 1 = 3 - 2
[1,1,0,1,0,0,1,0]
 => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
 => 2 = 4 - 2
[1,1,0,1,1,0,0,0]
 => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
[1,1,1,0,0,1,0,0]
 => 1 = 3 - 2
[1,1,1,0,1,0,0,0]
 => 1 = 3 - 2
[1,1,1,1,0,0,0,0]
 => 0 = 2 - 2

Description

The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. See [[http://www.findstat.org/DyckPaths/NakayamaAlgebras]]. The number of algebras where the statistic returns a value less than or equal to 1 might be given by the Motzkin numbers https://oeis.org/A001006.

Matching statistic: St000013
(load all 34 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [1,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 5 - 1

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St000058
(load all 44 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => 1 = 2 - 1
[1,0,1,0]
 => [1,2] => 1 = 2 - 1
[1,1,0,0]
 => [2,1] => 2 = 3 - 1
[1,0,1,0,1,0]
 => [1,2,3] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,3,2] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [2,1,3] => 2 = 3 - 1
[1,1,0,1,0,0]
 => [2,3,1] => 3 = 4 - 1
[1,1,1,0,0,0]
 => [3,2,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 2 = 3 - 1

Description

The order of a permutation.

$\operatorname{ord}(\pi)$ is given by the minimial

$k$ for which

$\pi^k$ is the identity permutation.

Matching statistic: St000308
(load all 275 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => 1 = 2 - 1
[1,0,1,0]
 => [2,1] => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => 2 = 3 - 1
[1,0,1,0,1,0]
 => [3,2,1] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [2,3,1] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [3,1,2] => 2 = 3 - 1
[1,1,0,1,0,0]
 => [2,1,3] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,2,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4 = 5 - 1

Description

The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices

$\{0,1,2,\ldots,n\}$ and root

$0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The statistic is given by the height of this tree. See also [[St000325]] for the width of this tree.

Matching statistic: St000328
(load all 14 compositions to match this statistic)

Mapping

Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000328: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [[]]
 => 1 = 2 - 1
[1,0,1,0]
 => [[],[]]
 => 2 = 3 - 1
[1,1,0,0]
 => [[[]]]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [[],[],[]]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [[],[[]]]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [[[]],[]]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [[[],[]]]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [[[[]]]]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 1 = 2 - 1

Description

The maximum number of child nodes in a tree.

Matching statistic: St000335
(load all 18 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000335: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [1,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 5 - 1

Description

The difference of lower and upper interactions. An ''upper interaction'' in a Dyck path is the occurrence of a factor

$0^k 1^k$ with

$k \geq 1$ (see [[St000331]]), and a ''lower interaction'' is the occurrence of a factor

$1^k 0^k$ with

$k \geq 1$ . In both cases,

$1$ denotes an up-step

$0$ denotes a a down-step.

The following 723 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000381The largest part of an integer composition. St000392The length of the longest run of ones in a binary word. St000808The number of up steps of the associated bargraph. St001058The breadth of the ordered tree. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St000160The multiplicity of the smallest part of a partition. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000094The depth of an ordered tree. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000983The length of the longest alternating subword. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001566The length of the longest arithmetic progression in a permutation. St000025The number of initial rises of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000054The first entry of the permutation. St000062The length of the longest increasing subsequence of the permutation. St000147The largest part of an integer partition. St000166The depth minus 1 of an ordered tree. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000527The width of the poset. St000542The number of left-to-right-minima of a permutation. St000628The balance of a binary word. St000662The staircase size of the code of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000846The maximal number of elements covering an element of a poset. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000982The length of the longest constant subword. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules 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. 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: 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. St001268The size of the largest ordinal summand in the poset. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001399The distinguishing number of a poset. St001555The order of a signed permutation. St001589The nesting number of a perfect matching. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001779The order of promotion on the set of linear extensions of a poset. St001809The index of the step at the first peak of maximal height in a Dyck path. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000209Maximum difference of elements in cycles. St000245The number of ascents of a permutation. St000306The bounce count of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000710The number of big deficiencies of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000873The aix statistic of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001090The number of pop-stack-sorts needed to sort a permutation. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000171The degree of the graph. St000172The Grundy number of a graph. St000203The number of external nodes of a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000271The chromatic index of a graph. St000325The width of the tree associated to a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000453The number of distinct Laplacian eigenvalues of a graph. St000470The number of runs in a permutation. St000528The height of a poset. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000638The number of up-down runs of a permutation. St000738The first entry in the last row of a standard tableau. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. 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

$c_{n-1}$ . St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. 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. St001118The acyclic chromatic index of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 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. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001315The dissociation number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001486The number of corners of the ribbon associated with an integer composition. 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001717The largest size of an interval in a poset. St001963The tree-depth of a graph. St000007The number of saliances of the permutation. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000080The rank of the poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000120The number of left tunnels of a Dyck path. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000225Difference between largest and smallest parts in a partition. St000240The number of indices that are not small excedances. St000297The number of leading ones in a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000314The number of left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000363The number of minimal vertex covers of a graph. St000382The first part of an integer composition. St000388The number of orbits of vertices of a graph under automorphisms. St000443The number of long tunnels of a Dyck path. St000469The distinguishing number of a graph. St000482The (zero)-forcing number of a graph. St000485The length of the longest cycle of a permutation. St000507The number of ascents of a standard tableau. St000619The number of cyclic descents of a permutation. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000676The number of odd rises of a Dyck path. St000703The number of deficiencies of a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000734The last entry in the first row of a standard tableau. St000740The last entry of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000778The metric dimension of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000845The maximal number of elements covered by an element in a poset. St000877The depth of the binary word interpreted as a path. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000971The smallest closer of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. 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. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001246The maximal difference between two consecutive entries of a permutation. St001270The bandwidth of a graph. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001298The number of repeated entries in the Lehmer code of a permutation. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001346The number of parking functions that give the same permutation. St001352The number of internal nodes in the modular decomposition of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001489The maximum of the number of descents and the number of inverse descents. St001494The Alon-Tarsi number of a graph. St001497The position of the largest weak excedence of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001570The minimal number of edges to add to make a graph Hamiltonian. St001580The acyclic chromatic number of a graph. St001590The crossing number of a perfect matching. St001674The number of vertices of the largest induced star graph in the graph. St001733The number of weak left to right maxima of a Dyck path. St001742The difference of the maximal and the minimal degree in a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001933The largest multiplicity of a part in an integer partition. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001962The proper pathwidth of a graph. St000024The number of double up and double down steps of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000154The sum of the descent bottoms of a permutation. St000214The number of adjacencies of a permutation. St000223The number of nestings in the permutation. St000237The number of small exceedances. St000272The treewidth of a graph. St000295The length of the border of a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000317The cycle descent number of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000338The number of pixed points of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000454The largest eigenvalue of a graph if it is integral. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000536The pathwidth of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000682The Grundy value of Welter's game on a binary word. St000711The number of big exceedences of a permutation. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000836The number of descents of distance 2 of a permutation. 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. 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. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001061The number of indices that are both descents and recoils of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001083The number of boxed occurrences of 132 in a permutation. St001096The size of the overlap set of a permutation. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 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. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of 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. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001358The largest degree of a regular subgraph of a graph. St001375The pancake length of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001578The minimal number of edges to add or remove to make a graph a line graph. St001638The book thickness of a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001777The number of weak descents in an integer composition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001826The maximal number of leaves on a vertex of a graph. St001864The number of excedances of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001949The rigidity index of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000472The sum of the ascent bottoms of a permutation. St000503The maximal difference between two elements in a common block. St000730The maximal arc length of a set partition. St000956The maximal displacement of a permutation. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St000061The number of nodes on the left branch of a binary tree. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000504The cardinality of the first block of a set partition. St000678The number of up steps after the last double rise of a Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000288The number of ones in a binary word. St000932The number of occurrences of the pattern UDU in a Dyck path. St000989The number of final rises of a permutation. St001435The number of missing boxes in the first row. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000386The number of factors DDU in a Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000022The number of fixed points of a permutation. St000035The number of left outer peaks of a permutation. St000259The diameter of a connected graph. St001488The number of corners of a skew partition. St000153The number of adjacent cycles of a permutation. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000215The number of adjacencies of a permutation, zero appended. St000260The radius of a connected graph. St000834The number of right outer peaks of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000891The number of distinct diagonal sums of a permutation matrix. St001782The order of rowmotion on the set of order ideals of a poset. St000907The number of maximal antichains of minimal length in a poset. St000912The number of maximal antichains in a poset. St000919The number of maximal left branches of a binary tree. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001645The pebbling number of a connected graph. St000884The number of isolated descents of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001624The breadth of a lattice. St000352The Elizalde-Pak rank of a permutation. St001394The genus of a permutation. St000806The semiperimeter of the associated bargraph. St001877Number of indecomposable injective modules with projective dimension 2. St001618The cardinality of the Frattini sublattice of a lattice. St000389The number of runs of ones of odd length in a binary word. St000568The hook number of a binary tree. St000659The number of rises of length at least 2 of a Dyck path. St000665The number of rafts of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001115The number of even descents of a permutation. St001182Number of indecomposable injective modules with codominant dimension at least two 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. St001875The number of simple modules with projective dimension at most 1. St000741The Colin de Verdière graph invariant. St000441The number of successions of a permutation. St000264The girth of a graph, which is not a tree. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001626The number of maximal proper sublattices of a lattice. St000390The number of runs of ones in a binary word. St001621The number of atoms of a lattice. St001863The number of weak excedances of a signed permutation. St000291The number of descents of a binary word. St000071The number of maximal chains in a poset. St000445The number of rises of length 1 of a Dyck path. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001893The flag descent of a signed permutation. St000534The number of 2-rises of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000201The number of leaf nodes in a binary tree. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000292The number of ascents of a binary word. St000456The monochromatic index of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001052The length of the exterior of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001896The number of right descents of a signed permutations. St000850The number of 1/2-balanced pairs in a poset. St001153The number of blocks with even minimum in a set partition. St001867The number of alignments of type EN of a signed permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000239The number of small weak excedances. St000254The nesting number of a set partition. St000502The number of successions of a set partitions. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001737The number of descents of type 2 in a permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St000023The number of inner peaks of a permutation. St000174The flush statistic of a semistandard tableau. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000663The number of right floats of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001712The number of natural descents of a standard Young tableau. St001728The number of invisible descents of a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001862The number of crossings of a signed permutation. St000422The energy of a graph, if it is integral. St000746The number of pairs with odd minimum in a perfect matching. St000820The number of compositions obtained by rotating the composition. St000904The maximal number of repetitions of an integer composition. St000918The 2-limited packing number of a graph. St000942The number of critical left to right maxima of the parking functions. St001060The distinguishing index of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001200The number of simple modules in

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

$A$ with minimal faithful projective-injective module

$eA$ . St001261The Castelnuovo-Mumford regularity of a graph. St001285The number of primes in the column sums of the two line notation of a permutation. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001471The magnitude of a Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001668The number of points of the poset minus the width of the poset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001806The upper middle entry of a permutation. St001902The number of potential covers of a poset. St000056The decomposition (or block) number of a permutation. St000238The number of indices that are not small weak excedances. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000488The number of cycles of a permutation of length at most 2. St000674The number of hills of a Dyck path. St000776The maximal multiplicity of an eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000925The number of topologically connected components of a set partition. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001152The number of pairs with even minimum in a perfect matching. St001188The number of simple modules

$S$ with grade

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

$A$ corresponding to the Dyck path. St001191Number of simple modules

$S$ with

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

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

$A$ with global dimension

$g$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001220The width of a permutation. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001405The number of bonds in a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001512The minimum rank of a graph. St001524The degree of symmetry of a binary word. St001665The number of pure excedances of a permutation. St001716The 1-improper chromatic number of a graph. St001732The number of peaks visible from the left. St001734The lettericity of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001769The reflection length of a signed permutation. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001792The arboricity of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001807The lower middle entry of a permutation. St001812The biclique partition number of a graph. St001820The size of the image of the pop stack sorting operator. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001866The nesting alignments of a signed permutation. St001874Lusztig's a-function for the symmetric group. St001884The number of borders of a binary word. St001928The number of non-overlapping descents in a permutation. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000004The major index of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000072The number of circled entries. St000083The number of left oriented leafs of a binary tree except the first one. St000173The segment statistic of a semistandard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000360The number of occurrences of the pattern 32-1. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000461The rix statistic of a permutation. St000640The rank of the largest boolean interval in a poset. St000650The number of 3-rises of a permutation. St000653The last descent of a permutation. St000735The last entry on the main diagonal of a standard tableau. St000761The number of ascents in an integer composition. St000779The tier of a permutation. St000794The mak of a permutation. St000872The number of very big descents of a permutation. St000961The shifted major index of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. 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. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001388The number of non-attacking neighbors of a permutation. St001423The number of distinct cubes in a binary word. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001781The interlacing number of a set partition. St001811The Castelnuovo-Mumford regularity of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001871The number of triconnected components of a graph. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St000438The position of the last up step in a Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St000981The length of the longest zigzag subpath. 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. St001019Sum of the projective dimensions of the simple modules in 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. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama 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. 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 St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000307The number of rowmotion orbits of a poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000677The standardized bi-alternating inversion number of a permutation. 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. St000893The number of distinct diagonal sums of an alternating sign matrix. St001623The number of doubly irreducible elements of a lattice. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000100The number of linear extensions of a poset. St000617The number of global maxima of a Dyck path. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000909The number of maximal chains of maximal size in a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001846The number of elements which do not have a complement in the lattice. St000075The orbit size of a standard tableau under promotion. St000736The last entry in the first row of a semistandard tableau. St001208The number of connected components of the quiver of

$A/T$ when

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

$A$ of

$K[x]/(x^n)$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St000089The absolute variation of a composition. St000090The variation of a composition. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000287The number of connected components of a graph. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000522The number of 1-protected nodes of a rooted tree. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000654The first descent of a permutation. St000739The first entry in the last row of a semistandard tableau. St000842The breadth of a permutation. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001637The number of (upper) dissectors of a poset. St001667The maximal size of a pair of weak twins for a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001926Sparre Andersen's position of the maximum of a signed permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000079The number of alternating sign matrices for a given Dyck path. St000298The order dimension or Dushnik-Miller dimension of a poset. St000315The number of isolated vertices of a graph. St000383The last part of an integer composition. St000519The largest length of a factor maximising the subword complexity. St000570The Edelman-Greene number of a permutation. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000839The largest opener of a set partition. St000862The number of parts of the shifted shape of a permutation. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000958The number of Bruhat factorizations of a permutation. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000988The orbit size of a permutation under Foata's bijection. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001162The minimum jump of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001344The neighbouring number of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001487The number of inner corners of a skew partition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001556The number of inversions of the third entry of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001735The number of permutations with the same set of runs. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between

$e_i J$ and

$e_j J$ (the radical of the indecomposable projective modules). St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000217The number of occurrences of the pattern 312 in a permutation. St000222The number of alignments in the permutation. St000230Sum of the minimal elements of the blocks of a set partition. St000232The number of crossings of a set partition. St000348The non-inversion sum of a binary word. St000355The number of occurrences of the pattern 21-3. St000407The number of occurrences of the pattern 2143 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000489The number of cycles of a permutation of length at most 3. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000516The number of stretching pairs of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000632The jump number of the poset. St000664The number of right ropes of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000954Number of times the corresponding LNakayama algebra has

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

$i>0$ . St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of

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

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001377The major index minus the number of inversions of a permutation. St001438The number of missing boxes of a skew partition. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001535The number of cyclic alignments of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001684The reduced word complexity of a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001841The number of inversions of a set partition. St001856The number of edges in the reduced word graph of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001911A descent variant minus the number of inversions. St001964The interval resolution global dimension of a poset.