Your data matches 408 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001232
(load all 236 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 6
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 7
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 6
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 7
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 8
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => 8
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => 10
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => 7
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => 10
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
 => 8
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 9
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => 10
[]
 => [1,0]
 => [1,0]
 => [1,0]
 => 0
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001213
(load all 2 compositions to match this statistic)
Mapping
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001213: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Values
[1,0]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 7 = 6 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 8 = 7 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 7 = 6 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 8 = 7 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 9 = 8 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 6 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 9 = 8 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 11 = 10 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 8 = 7 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => 11 = 10 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 9 = 8 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 10 = 9 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 11 = 10 + 1
[]
 => []
 => ?
 => ? = 0 + 1
Description
The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module.
Matching statistic: St000395
Mapping
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Values
[1,0]
 => [1,0]
 => [1,0]
 => [1,0]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 6
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 7
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 6
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 7
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 8
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 8
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 10
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 7
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 10
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 8
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 9
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 10
[]
 => []
 => []
 => ?
 => ? = 0
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St000029
(load all 2 compositions to match this statistic)
Mapping
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 82%
Values
[1,0]
 => [1,0]
 => [1,0]
 => [2,1] => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,3,1] => 2
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 3
[1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 4
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 4
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => 6
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => 7
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => 6
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => 7
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => 8
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ? = 6
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [2,3,7,5,6,1,4] => ? = 8
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => ? = 10
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [2,3,6,5,1,7,4] => ? = 7
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => ? = 10
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,7,4,1,6,3,5] => ? = 8
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => ? = 9
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => ? = 10
[]
 => []
 => []
 => [1] => 0
Description
The depth of a permutation. This is given by $$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$ The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$. Permutations with depth at most $1$ are called ''almost-increasing'' in [5].
Matching statistic: St000030
(load all 5 compositions to match this statistic)
Mapping
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 82%
Values
[1,0]
 => [1,0]
 => [2,1] => [2,1] => 1
[1,1,0,0]
 => [1,0,1,0]
 => [3,1,2] => [2,3,1] => 2
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => 3
[1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => [3,2,4,1] => 4
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [2,3,4,5,1] => 4
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [2,4,1,5,3] => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [4,2,3,5,1] => 6
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [3,4,1,5,6,2] => 7
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [2,3,5,1,6,4] => 6
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [2,5,4,1,6,3] => 7
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => [5,2,3,4,6,1] => 8
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 6
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [4,3,5,2,6,7,1] => ? = 8
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => [3,5,1,4,6,7,2] => ? = 10
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => [2,3,4,6,5,7,1] => ? = 7
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => [5,4,1,6,2,7,3] => ? = 10
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [2,3,6,1,4,7,5] => ? = 8
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => [2,6,4,5,1,7,3] => ? = 9
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => [6,2,3,4,5,7,1] => ? = 10
[]
 => []
 => [1] => [1] => 0
Description
The sum of the descent differences of a permutations. This statistic is given by $$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$ See [[St000111]] and [[St000154]] for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the ''drop'' of a permutation.
Matching statistic: St000841
Mapping
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000841: Perfect matchings ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 82%
Values
[1,0]
 => [1,0]
 => [1,0]
 => [(1,2)]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => 2
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 3
[1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 4
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => 6
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => 7
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => 6
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => 7
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => 8
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => ? = 6
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [(1,12),(2,5),(3,4),(6,11),(7,10),(8,9)]
 => ? = 8
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
 => ? = 10
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,5),(6,9),(7,8)]
 => ? = 7
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [(1,12),(2,3),(4,7),(5,6),(8,11),(9,10)]
 => ? = 10
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [(1,12),(2,11),(3,4),(5,6),(7,10),(8,9)]
 => ? = 8
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,11),(9,10)]
 => ? = 9
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => 10
[]
 => []
 => []
 => []
 => ? = 0
Description
The largest opener of a perfect matching. An opener (or left hand endpoint) of a perfect matching is a number that is matched with a larger number, which is then called a closer (or right hand endpoint).
Matching statistic: St001018
(load all 5 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001018: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 82%
Values
[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 8 = 7 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 7 = 6 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 8 = 7 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 9 = 8 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 6 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 8 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 10 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => ? = 7 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0,1,1,0,0]
 => ? = 10 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => ? = 8 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? = 9 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 10 + 1
[]
 => [1,0]
 => [1,0]
 => [1,0]
 => 1 = 0 + 1
Description
Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001065
(load all 4 compositions to match this statistic)
Mapping
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001065: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 82%
Values
[1,0]
 => [1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 4 = 2 + 2
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 5 = 3 + 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 6 = 4 + 2
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 6 = 4 + 2
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 7 = 5 + 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 8 = 6 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 7 = 5 + 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 9 = 7 + 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 8 = 6 + 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 9 = 7 + 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 10 = 8 + 2
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => ? = 6 + 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 8 + 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 10 + 2
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 2
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => ? = 10 + 2
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => ? = 8 + 2
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 9 + 2
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [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]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 10 + 2
[]
 => []
 => [1,0]
 => [1,0]
 => 2 = 0 + 2
Description
Number of indecomposable reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St000197
Mapping
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000197: Alternating sign matrices ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 73%
Values
[1,0]
 => [1,0]
 => [1,0]
 => [[1]]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [[0,1],[1,0]]
 => 2
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => 3
[1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 4
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => 6
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => 7
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => 6
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 7
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 8
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 6
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[1,0,0,-1,1,0],[0,0,0,1,0,0]]
 => ? = 8
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 10
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 7
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[1,0,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => ? = 10
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => ? = 8
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => ? = 9
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => ? = 10
[]
 => []
 => []
 => []
 => ? = 0
Description
The number of entries equal to positive one in the alternating sign matrix.
Matching statistic: St000957
(load all 7 compositions to match this statistic)
Mapping
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000957: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 73%
Values
[1,0]
 => [1,0]
 => [2,1] => [2,1] => 1
[1,1,0,0]
 => [1,0,1,0]
 => [3,1,2] => [3,2,1] => 2
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [4,2,1,3] => 3
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => [4,2,3,1] => 4
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5,4,2,1,3] => 4
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [5,2,3,1,4] => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5,2,3,4,1] => 6
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [6,4,2,1,3,5] => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [6,4,5,2,1,3] => 7
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [6,5,2,3,1,4] => 6
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => [6,2,3,4,1,5] => 7
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => [6,2,3,4,5,1] => 8
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => [7,6,4,2,1,3,5] => ? = 6
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,1,6,5,2,7,4] => [7,4,5,2,1,3,6] => ? = 8
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,7,5,6,2,4] => [7,4,5,6,2,1,3] => ? = 10
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [4,3,1,6,2,7,5] => [7,5,2,3,1,4,6] => ? = 7
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,3,1,7,6,2,5] => [7,5,6,2,3,1,4] => ? = 10
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,3,4,1,7,2,6] => [7,6,2,3,4,1,5] => ? = 8
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [6,3,4,5,1,7,2] => [7,2,3,4,5,1,6] => ? = 9
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [7,3,4,5,6,1,2] => [7,2,3,4,5,6,1] => ? = 10
[]
 => []
 => [1] => [1] => ? = 0
Description
The number of Bruhat lower covers of a permutation. This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$. This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.
The following 398 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001622The number of join-irreducible elements of a lattice. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St001717The largest size of an interval in a poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000189The number of elements in the poset. St000656The number of cuts of a poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000393The number of strictly increasing runs in a binary word. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000691The number of changes of a binary word. St000019The cardinality of the support of a permutation. St000538The number of even inversions of a permutation. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St001727The number of invisible inversions of a permutation. St000441The number of successions of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St001726The number of visible inversions of a permutation. St001875The number of simple modules with projective dimension at most 1. St000744The length of the path to the largest entry in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St001030Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001488The number of corners of a skew partition. St001645The pebbling number of a connected graph. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000766The number of inversions of an integer composition. St000912The number of maximal antichains in a poset. St001267The length of the Lyndon factorization of the binary word. St001462The number of factors of a standard tableaux under concatenation. St000806The semiperimeter of the associated bargraph. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000680The Grundy value for Hackendot on posets. St001343The dimension of the reduced incidence algebra of a poset. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001621The number of atoms of a lattice. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000632The jump number of the poset. St000670The reversal length of a permutation. St001424The number of distinct squares in a binary word. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000502The number of successions of a set partitions. St000519The largest length of a factor maximising the subword complexity. St000527The width of the poset. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000845The maximal number of elements covered by an element in a poset. St000922The minimal number such that all substrings of this length are unique. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001405The number of bonds in a permutation. St001549The number of restricted non-inversions between exceedances. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000021The number of descents of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000238The number of indices that are not small weak excedances. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000306The bounce count of a Dyck path. St000308The height of the tree associated to a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000482The (zero)-forcing number of a graph. St000539The number of odd inversions of a permutation. St000570The Edelman-Greene number of a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St000703The number of deficiencies of a permutation. St000778The metric dimension of a graph. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001270The bandwidth of a graph. St001282The number of graphs with the same chromatic polynomial. St001298The number of repeated entries in the Lehmer code of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001637The number of (upper) dissectors of a poset. St001644The dimension of a graph. St001668The number of points of the poset minus the width of the poset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001874Lusztig's a-function for the symmetric group. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000133The "bounce" of a permutation. St000173The segment statistic of a semistandard tableau. St000222The number of alignments in the permutation. St000235The number of indices that are not cyclical small weak excedances. St000236The number of cyclical small weak excedances. St000317The cycle descent number of a permutation. St000325The width of the tree associated to a permutation. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000470The number of runs in a permutation. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000516The number of stretching pairs of a permutation. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. 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. St000565The major index of a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. 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. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) 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. St000599The number of occurrences of the pattern {{1},{2,3}} such that (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. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (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. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is 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. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000636The hull number of a graph. St000638The number of up-down runs of a permutation. St000702The number of weak deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000836The number of descents of distance 2 of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000918The 2-limited packing number of a graph. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001286The annihilation number of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001388The number of non-attacking neighbors of a permutation. St001497The position of the largest weak excedence of a permutation. St001500The global dimension of magnitude 1 Nakayama algebras. St001512The minimum rank of a graph. St001513The number of nested exceedences of a permutation. St001516The number of cyclic bonds of a permutation. St001535The number of cyclic alignments of a permutation. St001537The number of cyclic crossings of a permutation. St001554The number of distinct nonempty subtrees of a binary tree. St001565The number of arithmetic progressions of length 2 in a permutation. St001566The length of the longest arithmetic progression in a permutation. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St001911A descent variant minus the number of inversions. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000080The rank of the poset. St000168The number of internal nodes of an ordered tree. St000174The flush statistic of a semistandard tableau. St000307The number of rowmotion orbits of a poset. St000327The number of cover relations in a poset. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000503The maximal difference between two elements in a common block. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000652The maximal difference between successive positions of a permutation. St000653The last descent of a permutation. St000710The number of big deficiencies of a permutation. St000747A variant of the major index of a set partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000956The maximal displacement of a permutation. St000989The number of final rises of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. 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. St001596The number of two-by-two squares inside a skew partition. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001728The number of invisible descents of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001812The biclique partition number of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001862The number of crossings of a signed permutation. St001869The maximum cut size of a graph. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001935The number of ascents in a parking function. St000004The major index of a permutation. St000015The number of peaks of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000064The number of one-box pattern of a permutation. St000089The absolute variation of a composition. St000090The variation of a composition. St000091The descent variation of a composition. St000105The number of blocks in the set partition. St000141The maximum drop size of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000166The depth minus 1 of an ordered tree. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000211The rank of the set partition. St000251The number of nonsingleton blocks of a set partition. St000254The nesting number of a set partition. St000287The number of connected components of a graph. St000305The inverse major index of a permutation. St000314The number of left-to-right-maxima of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000335The difference of lower and upper interactions. St000339The maf index of a permutation. St000352The Elizalde-Pak rank of a permutation. St000443The number of long tunnels of a Dyck path. St000446The disorder of 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. St000492The rob statistic of a set partition. St000493The los statistic of a set partition. St000498The lcs statistic of a set partition. St000499The rcb statistic of a set partition. St000504The cardinality of the first block of a set partition. St000505The biggest entry in the block containing the 1. St000522The number of 1-protected nodes of a rooted tree. St000546The number of global descents of a permutation. St000553The number of blocks of a graph. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000619The number of cyclic descents of a permutation. St000654The first descent of a permutation. St000662The staircase size of the code of a permutation. St000673The number of non-fixed points of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000740The last entry of a permutation. St000741The Colin de Verdière graph invariant. St000794The mak of a permutation. St000796The stat' of a permutation. St000798The makl of a permutation. St000822The Hadwiger number of the graph. St000823The number of unsplittable factors of the set partition. St000833The comajor index of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000881The number of short braid edges in the graph of braid moves of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000925The number of topologically connected components of a set partition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000942The number of critical left to right maxima of the parking functions. St000971The smallest closer of a set partition. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001114The number of odd descents of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001220The width of a permutation. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001517The length of a longest pair of twins in 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. St001530The depth of a Dyck path. St001557The number of inversions of the second entry of a permutation. St001638The book thickness of a graph. St001642The Prague dimension of a graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001729The number of visible descents of a permutation. St001769The reflection length of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001806The upper middle entry of a permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001926Sparre Andersen's position of the maximum of a signed permutation. St001928The number of non-overlapping descents in a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000094The depth of an ordered tree. St000104The number of facets in the order polytope of this poset. St000135The number of lucky cars of the parking function. St000151The number of facets in the chain polytope of the poset. St000383The last part of an integer composition. St000462The major index minus the number of excedences of a permutation. St000521The number of distinct subtrees of an ordered tree. St000542The number of left-to-right-minima of a permutation. St000650The number of 3-rises of a permutation. St000693The modular (standard) major index of a standard tableau. St000779The tier of a permutation. St000839The largest opener of a set partition. St000945The number of matchings in the dihedral orbit of a perfect matching. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001485The modular major index of a binary word. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001684The reduced word complexity of a permutation. St001697The shifted natural comajor index of a standard Young tableau. St001770The number of facets of a certain subword complex associated with the signed permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001807The lower middle entry of a permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000230Sum of the minimal elements of the blocks of a set partition. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000735The last entry on the main diagonal of a standard tableau.