Your data matches 558 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000052
(load all 29 compositions to match this statistic)
Mapping
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 0 = 1 - 1
[1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
Description
The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St001172
(load all 23 compositions to match this statistic)
Mapping
St001172: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 0 = 1 - 1
[1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
Description
The number of 1-rises at odd height of a Dyck path.
Matching statistic: St001066
(load all 46 compositions to match this statistic)
Mapping
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => 1
[1,0,1,0]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St001483
(load all 23 compositions to match this statistic)
Mapping
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => 1
[1,0,1,0]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => 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: St000065
(load all 27 compositions to match this statistic)
Mapping
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1]]
 => 0 = 1 - 1
[1,0,1,0]
 => [[1,0],[0,1]]
 => 0 = 1 - 1
[1,1,0,0]
 => [[0,1],[1,0]]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => 0 = 1 - 1
Description
The number of entries equal to -1 in an alternating sign matrix. The number of nonzero entries, [[St000890]] is twice this number plus the dimension of the matrix.
Matching statistic: St000496
(load all 8 compositions to match this statistic)
Mapping
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => {{1}}
 => 0 = 1 - 1
[1,0,1,0]
 => {{1},{2}}
 => 0 = 1 - 1
[1,1,0,0]
 => {{1,2}}
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => {{1,3},{2}}
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => {{1,2,3}}
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6}}
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4},{5,6}}
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3},{4,5},{6}}
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => {{1},{2},{3,4},{5},{6}}
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => {{1},{2},{3,4},{5,6}}
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => {{1},{2},{3,5},{4},{6}}
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => {{1},{2,3},{4},{5},{6}}
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => {{1},{2,3},{4},{5,6}}
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => {{1},{2,3},{4,5},{6}}
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => {{1},{2,4},{3},{5},{6}}
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6}}
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5,6}}
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6}}
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6}}
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => {{1,3},{2},{4},{5},{6}}
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6},{7}}
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => {{1},{2},{3},{4,5},{6},{7}}
 => 0 = 1 - 1
Description
The rcs statistic of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
Matching statistic: St000731
(load all 36 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => 0 = 1 - 1
[1,0,1,0]
 => [1,2] => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => 0 = 1 - 1
[1,1,0,1,0,0]
 => [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,4,2,5,6] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,6,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => 0 = 1 - 1
Description
The number of double exceedences of a permutation. A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St001033
(load all 20 compositions to match this statistic)
Mapping
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001033: 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
[1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => 0 = 1 - 1
Description
The normalized area of the parallelogram polyomino associated with the Dyck path. The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path. The area itself is equidistributed with [[St001034]] and with [[St000395]].
Matching statistic: St001167
(load all 19 compositions to match this statistic)
Mapping
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
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
[1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
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 19 compositions to match this statistic)
Mapping
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
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
[1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
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).
The following 548 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001781The interlacing number of a set partition. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St000058The order of a permutation. St000071The number of maximal chains in a poset. St000110The number of permutations less than or equal to a permutation in left weak order. St000451The length of the longest pattern of the form k 1 2. St000909The number of maximal chains of maximal size in a poset. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000141The maximum drop size of a permutation. St000209Maximum difference of elements in cycles. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000356The number of occurrences of the pattern 13-2. 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 $3$. St000374The number of exclusive right-to-left minima of a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000463The number of admissible inversions of a permutation. St000534The number of 2-rises of a permutation. St000552The number of cut vertices of a graph. St000670The reversal length of a permutation. St000703The number of deficiencies of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St001090The number of pop-stack-sorts needed to sort a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001308The number of induced paths on three vertices in a graph. St001323The independence gap of a graph. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001584The area statistic between a Dyck path and its bounce path. 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. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St000013The height of a Dyck path. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000345The number of refinements of a partition. St000363The number of minimal vertex covers of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000444The length of the maximal rise of a Dyck path. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000470The number of runs in a permutation. 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. St000883The number of longest increasing subsequences of a permutation. St000935The number of ordered refinements of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001304The number of maximally independent sets of vertices 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. St001389The number of partitions of the same length below the given integer partition. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001725The harmonious chromatic number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. 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. St000057The Shynar inversion number of a standard tableau. St000081The number of edges of a graph. 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. St000171The degree of the graph. St000185The weighted size of a partition. St000204The number of internal nodes of a binary tree. St000211The rank of the set partition. St000214The number of adjacencies of a permutation. St000223The number of nestings in the permutation. St000225Difference between largest and smallest parts in a partition. St000233The number of nestings of a set partition. St000237The number of small exceedances. St000246The number of non-inversions of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000352The Elizalde-Pak rank of a permutation. St000355The number of occurrences of the pattern 21-3. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000442The maximal area to the right of an up step of a Dyck path. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St000651The maximal size of a rise in a permutation. St000662The staircase size of the code of a permutation. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000884The number of isolated descents of a permutation. 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. St001083The number of boxed occurrences of 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001120The length of a longest path in a graph. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001214The aft of an integer partition. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. 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. 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. 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. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001411The number of patterns 321 or 3412 in 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. St001638The book thickness of a graph. St001665The number of pure excedances of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001826The maximal number of leaves on a vertex of a graph. St001842The major index of a set partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001928The number of non-overlapping descents in a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) 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. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 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. St000485The length of the longest cycle of a permutation. St000652The maximal difference between successive positions of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001246The maximal difference between two consecutive entries of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. 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. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) 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. 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. St000795The mad of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000011The number of touch points (or returns) of a Dyck path. St000100The number of linear extensions of a poset. St000456The monochromatic index of a connected graph. St000619The number of cyclic descents of a permutation. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000983The length of the longest alternating subword. St001062The maximal size of a block of a set partition. St001128The exponens consonantiae of a partition. St001313The number of Dyck paths above the lattice path given by a binary word. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001642The Prague dimension of a graph. St000232The number of crossings of a set partition. St000248The number of anti-singletons of a set partition. St000354The number of recoils of a permutation. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000539The number of odd inversions of a permutation. St000565The major index of a set partition. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000653The last descent of a permutation. St000682The Grundy value of Welter's game on a binary word. St000711The number of big exceedences of a permutation. St000728The dimension of a set partition. St000732The number of double deficiencies of a permutation. St000794The mak of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000836The number of descents of distance 2 of a permutation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001061The number of indices that are both descents and recoils of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001689The number of celebrities in a graph. St000886The number of permutations with the same antidiagonal sums. St000002The number of occurrences of the pattern 123 in a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000710The number of big deficiencies of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000220The number of occurrences of the pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000143The largest repeated part of a partition. St000367The number of simsun double descents of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000932The number of occurrences of the pattern UDU in a Dyck path. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St000039The number of crossings of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. 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. St000255The number of reduced Kogan faces with the permutation as type. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000316The number of non-left-to-right-maxima of a permutation. St000317The cycle descent number of a permutation. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. 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. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St000086The number of subgraphs. St000272The treewidth of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000325The width of the tree associated to a permutation. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000536The pathwidth of a graph. St000730The maximal arc length of a set partition. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001268The size of the largest ordinal summand in the poset. St001271The competition number of a graph. St001277The degeneracy of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001779The order of promotion on the set of linear extensions of a poset. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000004The major index of a permutation. St000021The number of descents of a permutation. St000067The inversion number of the alternating sign matrix. St000095The number of triangles of a graph. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000156The Denert index of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000332The positive inversions of an alternating sign matrix. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra 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 $c_{n-1}$. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001117The game chromatic index of a graph. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001377The major index minus the number of inversions of a permutation. St001397Number of pairs of incomparable elements in a finite poset. St001428The number of B-inversions of a signed permutation. 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. St001580The acyclic chromatic number of a graph. St001649The length of a longest trail in a graph. St001742The difference of the maximal and the minimal degree in a graph. 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. St000216The absolute length of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001082The number of boxed occurrences of 123 in a permutation. St000526The number of posets with combinatorially isomorphic order polytopes. St001220The width of a permutation. St001346The number of parking functions that give the same permutation. St001959The product of the heights of the peaks of a Dyck path. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000849The number of 1/3-balanced pairs in a poset. St000961The shifted major index of a permutation. St001130The number of two successive successions in a permutation. St001480The number of simple summands of the module J^2/J^3. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001571The Cartan determinant of the integer partition. St001933The largest multiplicity of a part in an integer partition. St000219The number of occurrences of the pattern 231 in a permutation. St000392The length of the longest run of ones in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St000439The position of the first down step of a Dyck path. 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. St000848The balance constant multiplied with the number of linear extensions of a poset. St000450The number of edges minus the number of vertices plus 2 of a graph. St001549The number of restricted non-inversions between exceedances. St001812The biclique partition number of a graph. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000649The number of 3-excedences of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001513The number of nested exceedences of a permutation. St001530The depth of a Dyck path. St000215The number of adjacencies of a permutation, zero appended. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001330The hat guessing number of a graph. St000150The floored half-sum of the multiplicities of a partition. St000257The number of distinct parts of a partition that occur at least twice. St001624The breadth of a lattice. St000624The normalized sum of the minimal distances to a greater element. St001877Number of indecomposable injective modules with projective dimension 2. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000779The tier of a permutation. St001399The distinguishing number of a poset. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000516The number of stretching pairs of a permutation. St001684The reduced word complexity of a permutation. St000570The Edelman-Greene number of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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. St001298The number of repeated entries in the Lehmer code of a permutation. St001555The order of a signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St000360The number of occurrences of the pattern 32-1. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000822The Hadwiger number of the graph. St001186Number of simple modules with grade at least 3 in 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. 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. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001769The reflection length of a signed permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001569The maximal modular displacement of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001867The number of alignments of type EN of a signed permutation. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001846The number of elements which do not have a complement in the lattice. St001095The number of non-isomorphic posets with precisely one further covering relation. St001720The minimal length of a chain of small intervals in a lattice. St001868The number of alignments of type NE of a signed permutation. St001115The number of even descents of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000667The greatest common divisor of the parts of the partition. St001820The size of the image of the pop stack sorting operator. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001964The interval resolution global dimension of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000074The number of special entries. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001550The number of inversions between exceedances where the greater exceedance is linked. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001394The genus of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St001118The acyclic chromatic index of a graph. St000068The number of minimal elements in a poset. 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. St001890The maximum magnitude of the Möbius function of a poset. St000007The number of saliances of the permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000264The girth of a graph, which is not a tree. St001875The number of simple modules with projective dimension at most 1. St000842The breadth of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in 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. 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 $eA$. St000295The length of the border of a binary word. St000706The product of the factorials of the multiplicities of an integer partition. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St001613The binary logarithm of the size of the center of a lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001881The number of factors of a lattice as a Cartesian product of lattices. St000929The constant term of the character polynomial of an integer partition. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001616The number of neutral elements in a lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000338The number of pixed points of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001705The number of occurrences of the pattern 2413 in a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000239The number of small weak excedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000535The rank-width of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000805The number of peaks of the associated bargraph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000937The number of positive values of the symmetric group character corresponding to the partition. St000942The number of critical left to right maxima of the parking functions. 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 $K[x]/(x^n)$. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001413Half the length of the longest even length palindromic prefix of a binary word. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001735The number of permutations with the same set of runs. St001743The discrepancy of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001904The length of the initial strictly increasing segment of a parking function. St001946The number of descents in a parking function. St000090The variation of a composition. St000091The descent variation of a composition. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000308The height of the tree associated to a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000478Another weight of a partition according to Alladi. St000487The length of the shortest cycle of a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000542The number of left-to-right-minima of a permutation. St000562The number of internal points of a set partition. St000664The number of right ropes of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000709The number of occurrences of 14-2-3 or 14-3-2. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001114The number of odd descents of a permutation. St001153The number of blocks with even minimum in a set partition. 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. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001353The number of prime nodes in the modular decomposition of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001434The number of negative sum pairs of a signed permutation. St001470The cyclic holeyness of a permutation. St001537The number of cyclic crossings of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001691The number of kings in a graph. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001797The number of overfull subgraphs of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001857The number of edges in the reduced word graph of a signed permutation. St001884The number of borders of a binary word. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size in a poset. St000914The sum of the values of the Möbius function of a poset. St001487The number of inner corners of a skew partition. 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. St000298The order dimension or Dushnik-Miller dimension of a poset. St000486The number of cycles of length at least 3 of a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000872The number of very big descents of a permutation. St000907The number of maximal antichains of minimal length in a poset. 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. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001260The permanent of an alternating sign matrix. St000894The trace of an alternating sign matrix. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.