- FindStat

Your data matches 658 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001022
(load all 14 compositions to match this statistic)

Mapping

St001022: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 0
[1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => 1
[1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => 0
[1,1,0,1,0,0]
 => 0
[1,1,1,0,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => 0
[1,1,1,1,0,0,1,0,0,0]
 => 0
[1,1,1,1,0,1,0,0,0,0]
 => 0
[1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,1,0,0]
 => 0
[1,0,1,1,1,1,0,0,1,0,0,0]
 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,1,1,0,0,1,0,0]
 => 0
[1,1,0,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0,1,0]
 => 0
[1,1,0,1,1,1,0,0,0,1,0,0]
 => 0

Description

Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001167
(load all 14 compositions to match this statistic)

Mapping

St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 0
[1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => 1
[1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => 0
[1,1,0,1,0,0]
 => 0
[1,1,1,0,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => 0
[1,1,1,1,0,0,1,0,0,0]
 => 0
[1,1,1,1,0,1,0,0,0,0]
 => 0
[1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,1,0,0]
 => 0
[1,0,1,1,1,1,0,0,1,0,0,0]
 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,1,1,0,0,1,0,0]
 => 0
[1,1,0,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0,1,0]
 => 0
[1,1,0,1,1,1,0,0,0,1,0,0]
 => 0

Description

The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. The top of a module is the cokernel of the inclusion of the radical of the module into the module. For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.

Matching statistic: St001253
(load all 14 compositions to match this statistic)

Mapping

St001253: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 0
[1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => 1
[1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => 0
[1,1,0,1,0,0]
 => 0
[1,1,1,0,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => 0
[1,1,1,1,0,0,1,0,0,0]
 => 0
[1,1,1,1,0,1,0,0,0,0]
 => 0
[1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,1,0,0]
 => 0
[1,0,1,1,1,1,0,0,1,0,0,0]
 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,1,1,0,0,1,0,0]
 => 0
[1,1,0,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0,1,0]
 => 0
[1,1,0,1,1,1,0,0,0,1,0,0]
 => 0

Description

The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. For the first 196 values the statistic coincides also with the number of fixed points of

$\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details. The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).

Matching statistic: St001483
(load all 17 compositions to match this statistic)

Mapping

St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1

Description

The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.

Matching statistic: St001503
(load all 16 compositions to match this statistic)

Mapping

St001503: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1

Description

The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra.

Matching statistic: St000039
(load all 20 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of crossings of a permutation. A crossing of a permutation

$\pi$ is given by a pair

$(i,j)$ such that either

$i < j \leq \pi(i) \leq \pi(j)$ or

$\pi(i) < \pi(j) < i < j$ . Pictorially, the diagram of a permutation is obtained by writing the numbers from

$1$ to

$n$ in this order on a line, and connecting

$i$ and

$\pi(i)$ with an arc above the line if

$i\leq\pi(i)$ and with an arc below the line if

$i > \pi(i)$ . Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.

Matching statistic: St000436
(load all 13 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000436: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the pattern 231 or of the pattern 321 in a permutation.

Matching statistic: St000660
(load all 11 compositions to match this statistic)

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,1,0,0]
 => 0
[1,1,0,0]
 => [1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 0

Description

The number of rises of length at least 3 of a Dyck path. The number of Dyck paths without such rises are counted by the Motzkin numbers [1].

Matching statistic: St000661
(load all 9 compositions to match this statistic)

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,1,0,0]
 => 0
[1,1,0,0]
 => [1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 0

Description

The number of rises of length 3 of a Dyck path.

Matching statistic: St000710
(load all 16 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000710: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of big deficiencies of a permutation. A big deficiency of a permutation

$\pi$ is an index

$i$ such that

$i - \pi(i) > 1$ . This statistic is equidistributed with any of the numbers of big exceedences, big descents and big ascents.

The following 648 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000731The number of double exceedences of a permutation. St000779The tier of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000920The logarithmic height of a Dyck path. St000002The number of occurrences of the pattern 123 in a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000052The number of valleys of a Dyck path not on the x-axis. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000217The number of occurrences of the pattern 312 in a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000223The number of nestings in the permutation. St000317The cycle descent number of a permutation. St000356The number of occurrences of the pattern 13-2. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. 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. St000372The number of mid points of increasing 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

$3$ . St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000632The jump number of the poset. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001082The number of boxed occurrences of 123 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. 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. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001394The genus of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001411The number of patterns 321 or 3412 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. 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. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000071The number of maximal chains in a poset. St000078The number of alternating sign matrices whose left key is the permutation. St000100The number of linear extensions of a poset. St000124The cardinality of the preimage of the Simion-Schmidt map. St000201The number of leaf nodes in a binary tree. St000255The number of reduced Kogan faces with the permutation as type. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000396The register function (or Horton-Strahler number) of a binary tree. St000527The width of the poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. 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. St001268The size of the largest ordinal summand in the poset. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001399The distinguishing number of a poset. St001597The Frobenius rank of a skew partition. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000012The area of a Dyck path. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000081The number of edges of a graph. St000095The number of triangles of a graph. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000171The degree of the graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000232The number of crossings of a set partition. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000271The chromatic index of a graph. St000272The treewidth of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000354The number of recoils of a permutation. St000355The number of occurrences of the pattern 21-3. St000359The number of occurrences of the pattern 23-1. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000386The number of factors DDU in a Dyck path. St000387The matching number of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000475The number of parts equal to 1 in a partition. St000481The number of upper covers of a partition in dominance order. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000523The number of 2-protected nodes of a rooted tree. St000534The number of 2-rises of a permutation. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000539The number of odd inversions of a permutation. St000552The number of cut vertices of a graph. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000572The dimension exponent of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. 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. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. St000674The number of hills of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000703The number of deficiencies of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000884The number of isolated descents of a permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000984The number of boxes below precisely one peak. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001071The beta invariant of the graph. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001092The number of distinct even parts of a partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001214The aft of an integer partition. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001341The number of edges in the center of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001469The holeyness of a permutation. St001479The number of bridges of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001512The minimum rank of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001584The area statistic between a Dyck path and its bounce path. St001587Half of the largest even part of an integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001618The cardinality of the Frattini sublattice of a lattice. St001638The book thickness of a graph. St001644The dimension of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001649The length of a longest trail in a graph. St001665The number of pure excedances of a permutation. St001718The number of non-empty open intervals in a poset. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001736The total number of cycles in a graph. St001737The number of descents of type 2 in a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001743The discrepancy of a graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001781The interlacing number of a set partition. St001792The arboricity of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001869The maximum cut size of a graph. St001874Lusztig's a-function for the symmetric group. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001928The number of non-overlapping descents in a permutation. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001962The proper pathwidth of a graph. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000068The number of minimal elements in a poset. St000079The number of alternating sign matrices for a given Dyck path. St000086The number of subgraphs. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000099The number of valleys of a permutation, including the boundary. St000146The Andrews-Garvan crank of a partition. St000159The number of distinct parts of the integer partition. St000172The Grundy number of a graph. St000183The side length of the Durfee square of an integer partition. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000299The number of nonisomorphic vertex-induced subtrees. St000325The width of the tree associated to a permutation. St000343The number of spanning subgraphs of a graph. St000346The number of coarsenings of a partition. St000363The number of minimal vertex covers of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000450The number of edges minus the number of vertices plus 2 of a graph. St000451The length of the longest pattern of the form k 1 2. St000453The number of distinct Laplacian eigenvalues of a graph. St000456The monochromatic index of a connected graph. St000468The Hosoya index of a graph. St000470The number of runs in a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000482The (zero)-forcing number of a graph. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000544The cop number of a graph. 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. St000678The number of up steps after the last double rise of a Dyck path. St000679The pruning number of an ordered tree. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000701The protection number of a binary tree. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000758The length of the longest staircase fitting into an integer composition. St000759The smallest missing part in an integer partition. St000778The metric dimension of a graph. St000783The side length of the largest staircase partition fitting into a partition. St000785The number of distinct colouring schemes of a graph. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000822The Hadwiger number of the graph. St000862The number of parts of the shifted shape of a permutation. St000886The number of permutations with the same antidiagonal sums. St000972The composition number of a graph. St001007Number of simple modules with projective dimension 1 in 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. St001029The size of the core of a graph. St001063Numbers of 3-torsionfree 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. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001196The global dimension of

$A$ minus the global dimension of

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

$eA$ . St001246The maximal difference between two consecutive entries of a permutation. St001261The Castelnuovo-Mumford regularity of a graph. St001271The competition number of a graph. St001280The number of parts of an integer partition that are at least two. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001432The order dimension of the partition. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001484The number of singletons of an integer partition. St001494The Alon-Tarsi number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001624The breadth of a lattice. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001732The number of peaks visible from the left. St001734The lettericity of a graph. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001883The mutual visibility number of a graph. 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000397The Strahler number of a rooted tree. St000439The position of the first down step of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . St001642The Prague dimension of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001746The coalition number of a graph. St000219The number of occurrences of the pattern 231 in a permutation. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000741The Colin de Verdière graph invariant. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000264The girth of a graph, which is not a tree. 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. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. 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. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000883The number of longest increasing subsequences of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000649The number of 3-excedences of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001570The minimal number of edges to add to make a graph Hamiltonian. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. 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. St000842The breadth of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000993The multiplicity of the largest part of an integer partition. St001060The distinguishing index of a graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000567The sum of the products of all pairs of parts. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001162The minimum jump of a permutation. 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. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000007The number of saliances of the permutation. St000237The number of small exceedances. St000247The number of singleton blocks of a set partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000461The rix statistic of a permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001811The Castelnuovo-Mumford regularity of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000221The number of strong fixed points of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000338The number of pixed points of a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000872The number of very big descents of a permutation. St000951The dimension of

$Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001193The dimension of

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

$A$ such that

$eA$ is a minimal faithful projective-injective module. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001705The number of occurrences of the pattern 2413 in a permutation. St001835The number of occurrences of a 231 pattern in the restricted growth word of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. 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). St001960The number of descents of a permutation minus one if its first entry is not one. St000105The number of blocks in the set partition. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000570The Edelman-Greene number of a permutation. St000654The first descent of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000864The number of circled entries of the shifted recording tableau of a permutation. St000873The aix statistic of a permutation. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001188The number of simple modules

$S$ with grade

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

$A$ corresponding to the Dyck path. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001481The minimal height of a peak of a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001531Number of partial orders contained in the poset determined by the Dyck path. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001959The product of the heights of the peaks of a 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. St001239The largest vector space dimension of the double dual of a simple module 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001200The number of simple modules in

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

$A$ with minimal faithful projective-injective module

$eA$ . St000807The sum of the heights of the valleys of the associated bargraph. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St000260The radius of a connected graph. St000259The diameter of a connected graph. St001651The Frankl number of a lattice. St000914The sum of the values of the Möbius function of a poset. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001890The maximum magnitude of the Möbius function of a poset. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001866The nesting alignments of a signed permutation. St001770The number of facets of a certain subword complex associated with the signed permutation. St001964The interval resolution global dimension of a poset. St001545The second Elser number of a connected graph. St001535The number of cyclic alignments of a permutation. St001846The number of elements which do not have a complement in the lattice. St000031The number of cycles in the cycle decomposition of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001423The number of distinct cubes in a binary word. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000805The number of peaks of the associated bargraph. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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

$K[x]/(x^n)$ . St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St000058The order of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001115The number of even descents of a permutation. St000834The number of right outer peaks of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001490The number of connected components of a skew partition. St000153The number of adjacent cycles of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000405The number of occurrences of the pattern 1324 in a permutation. St000906The length of the shortest maximal chain in a poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001625The Möbius invariant of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001613The binary logarithm of the size of the center of a lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001857The number of edges in the reduced word graph of a signed permutation. St001344The neighbouring number of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000017The number of inversions of a standard tableau. St000091The descent variation of a composition. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000241The number of cyclical small excedances. St000370The genus of a graph. St000406The number of occurrences of the pattern 3241 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000650The number of 3-rises of a permutation. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. 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. 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. St001309The number of four-cliques in a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001470The cyclic holeyness of a permutation. St001715The number of non-records in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001856The number of edges in the reduced word graph of a permutation. St001867The number of alignments of type EN of a signed permutation. St001871The number of triconnected components of a graph. St001948The number of augmented double ascents of a permutation. St000239The number of small weak excedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000353The number of inner valleys of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001220The width of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001768The number of reduced words of a signed permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000092The number of outer peaks of a permutation. St000236The number of cyclical small weak 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. St000308The height of the tree associated to a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000504The cardinality of the first block of a set partition. St000636The hull number of a graph. St000742The number of big ascents of a permutation after prepending zero. St001062The maximal size of a block of a set partition. St001096The size of the overlap set of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001471The magnitude of a Dyck path. St001654The monophonic hull number of a graph. St000717The number of ordinal summands of a poset. St000891The number of distinct diagonal sums of a permutation matrix. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001301The first Betti number of the order complex associated with the poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules

$S$ with

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

$A$ of the poset. St001845The number of join irreducibles minus the rank of a lattice. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset.